From stdpp Require Import prelude.
From Coq Require Import FinFun Lia.
From VLSM.Lib Require Import Preamble ListExtras StdppListSet FinExtras.
From VLSM Require Import Core.VLSM Core.VLSMProjections Core.Composition Core.Equivocation.
From VLSM Require Import Core.Equivocation.NoEquivocation Core.Equivocation.FullNode Core.Equivocation.FixedSetEquivocation.
From VLSM Require Import Core.SubProjectionTraces Core.ProjectionTraces.
From VLSM Require Import Core.Equivocators.Common Core.Equivocators.Projections.
From VLSM Require Import Core.Equivocators.MessageProperties.
From VLSM Require Import Core.Equivocators.Composition.Common.
From VLSM Require Import Core.Equivocators.Composition.Projections.

VLSM Equivocators Fixed Equivocation

Section equivocators_fixed_equivocations_vlsm.

Context
  {message : Type}
  {index : Type}
  {IndEqDec : EqDecision index}
  (IM : index -> VLSM message)
  (Hbs : forall i : index, HasBeenSentCapability (IM i))
  (equivocator_IM := equivocator_IM IM)
  (index_listing : list index)
  (finite_index : Listing index_listing)
  (equivocating : set index)
  .

Definition of fixed-equivocation for states of the composition of equivocators.

Definition state_has_fixed_equivocation
  (s : composite_state equivocator_IM)
  : Prop
  :=
  (equivocating_indices IM index_listing s) ⊆ equivocating.

Definition equivocators_fixed_equivocations_constraint
  (l : composite_label equivocator_IM)
  (som : composite_state equivocator_IM * option message)
  (som' := composite_transition equivocator_IM l som)
  : Prop
  := equivocators_no_equivocations_constraint IM Hbs l som
  /\ state_has_fixed_equivocation (fst som').

If the starting state_has_fixed_equivocation and the transition is made on an equivocating index, then the resulting state_has_fixed_equivocation.

Lemma equivocating_transition_preserves_fixed_equivocation l s om s' om'
  (Ht : composite_transition equivocator_IM l (s, om) = (s', om'))
  (Hs : state_has_fixed_equivocation s)
  (Heqv : projT1 l ∈ equivocating)
  : state_has_fixed_equivocation s'.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
Heqv: projT1 l ∈ equivocating
state_has_fixed_equivocation s'
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
Heqv: projT1 l ∈ equivocating
state_has_fixed_equivocation s'
  intros i Hi.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
Heqv: projT1 l ∈ equivocating
i: index
Hi: i ∈ equivocating_indices IM index_listing s'
i ∈ equivocating
  destruct (decide (i = projT1 l))
  ; [subst; assumption|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
Heqv: projT1 l ∈ equivocating
i: index
Hi: i ∈ equivocating_indices IM index_listing s'
n: i ≠ projT1 l
i ∈ equivocating
  destruct l; cbn in *.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
x: index
v: EquivocatorLabel (IM x)
s: _composite_state equivocator_IM
om: option message
s': _composite_state equivocator_IM
om': option message
Ht: (let (si', om') :=
   equivocator_transition (IM x) v (s x, om) in
 (state_update equivocator_IM s x si', om')) =
(s', om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: i ∈ equivocating_indices IM index_listing s'
n: i ≠ x
i ∈ equivocating
  destruct (equivocator_transition _ _ _).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
x: index
v: EquivocatorLabel (IM x)
s: _composite_state equivocator_IM
om: option message
s': _composite_state equivocator_IM
om': option message
e: equivocator_state (IM x)
o: option message
Ht: (state_update equivocator_IM s x e, o) =
(s', om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: i ∈ equivocating_indices IM index_listing s'
n: i ≠ x
i ∈ equivocating
  inversion Ht.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
x: index
v: EquivocatorLabel (IM x)
s: _composite_state equivocator_IM
om: option message
s': _composite_state equivocator_IM
om': option message
e: equivocator_state (IM x)
o: option message
Ht: (state_update equivocator_IM s x e, o) =
(s', om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: i ∈ equivocating_indices IM index_listing s'
n: i ≠ x
H0: state_update equivocator_IM s x e = s'
H1: o = om'
i ∈ equivocating subst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
x: index
v: EquivocatorLabel (IM x)
s: _composite_state equivocator_IM
om, om': option message
e: equivocator_state (IM x)
Ht: (state_update equivocator_IM s x e, om') =
(state_update equivocator_IM s x e, om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: i
∈ equivocating_indices IM index_listing
    (state_update equivocator_IM s x e)
n: i ≠ x
i ∈ equivocating
  apply elem_of_list_filter, proj1 in Hi.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
x: index
v: EquivocatorLabel (IM x)
s: _composite_state equivocator_IM
om, om': option message
e: equivocator_state (IM x)
Ht: (state_update equivocator_IM s x e, om') =
(state_update equivocator_IM s x e, om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: is_equivocating_state (IM i)
  (state_update equivocator_IM s x e i)
n: i ≠ x
i ∈ equivocating
  rewrite state_update_neq in Hi by congruence.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
x: index
v: EquivocatorLabel (IM x)
s: _composite_state equivocator_IM
om, om': option message
e: equivocator_state (IM x)
Ht: (state_update equivocator_IM s x e, om') =
(state_update equivocator_IM s x e, om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: is_equivocating_state (IM i) (s i)
n: i ≠ x
i ∈ equivocating
  apply Hs.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
x: index
v: EquivocatorLabel (IM x)
s: _composite_state equivocator_IM
om, om': option message
e: equivocator_state (IM x)
Ht: (state_update equivocator_IM s x e, om') =
(state_update equivocator_IM s x e, om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: is_equivocating_state (IM i) (s i)
n: i ≠ x
i ∈ equivocating_indices IM index_listing s
  apply elem_of_list_filter.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
x: index
v: EquivocatorLabel (IM x)
s: _composite_state equivocator_IM
om, om': option message
e: equivocator_state (IM x)
Ht: (state_update equivocator_IM s x e, om') =
(state_update equivocator_IM s x e, om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: is_equivocating_state (IM i) (s i)
n: i ≠ x
is_equivocating_state (IM i) (s i) ∧ i ∈ index_listing
  split; [assumption|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
x: index
v: EquivocatorLabel (IM x)
s: _composite_state equivocator_IM
om, om': option message
e: equivocator_state (IM x)
Ht: (state_update equivocator_IM s x e, om') =
(state_update equivocator_IM s x e, om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: is_equivocating_state (IM i) (s i)
n: i ≠ x
i ∈ index_listing
  apply elem_of_list_In.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
x: index
v: EquivocatorLabel (IM x)
s: _composite_state equivocator_IM
om, om': option message
e: equivocator_state (IM x)
Ht: (state_update equivocator_IM s x e, om') =
(state_update equivocator_IM s x e, om')
Hs: state_has_fixed_equivocation s
Heqv: x ∈ equivocating
i: index
Hi: is_equivocating_state (IM i) (s i)
n: i ≠ x
In i index_listing apply finite_index.
Qed.

If the starting state_has_fixed_equivocation and the transition is made on the original copy, then the resulting state_has_fixed_equivocation.

Lemma zero_descriptor_transition_preserves_fixed_equivocation l s om s' om'
  (Ht : composite_transition equivocator_IM l (s, om) = (s', om'))
  (Hs : state_has_fixed_equivocation s)
  li
  (Hzero : projT2 l = ContinueWith 0 li)
  : state_has_fixed_equivocation s'.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
state_has_fixed_equivocation s'
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
state_has_fixed_equivocation s'
  intros i Hi.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
i: index
Hi: i ∈ equivocating_indices IM index_listing s'
i ∈ equivocating apply Hs.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
i: index
Hi: i ∈ equivocating_indices IM index_listing s'
i ∈ equivocating_indices IM index_listing s
  apply elem_of_list_filter, proj1 in Hi.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
i ∈ equivocating_indices IM index_listing s
  apply elem_of_list_filter.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
is_equivocating_state (IM i) (s i) ∧ i ∈ index_listing
  split; [|apply elem_of_list_In; apply finite_index].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: composite_transition equivocator_IM l (s, om) =
(s', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
is_equivocating_state (IM i) (s i)
  cbn in Ht.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label equivocator_IM
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: (let (i, li) := l in
 let (si', om') :=
   equivocator_transition (IM i) li (s i, om) in
 (state_update equivocator_IM s i si', om')) =
(s', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 l))
Hzero: projT2 l = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
is_equivocating_state (IM i) (s i)
  destruct l as (eqv, leqv).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
eqv: index
leqv: vlabel (equivocator_IM eqv)
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
Ht: (let (si', om') :=
   equivocator_transition 
     (IM eqv) leqv (s eqv, om) in
 (state_update equivocator_IM s eqv si', om')) =
(s', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
is_equivocating_state (IM i) (s i)
  destruct (equivocator_transition _ _ _) as (si', _om') eqn:Hti.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
eqv: index
leqv: vlabel (equivocator_IM eqv)
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
si': equivocator_state (IM eqv)
_om': option message
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', _om')
Ht: (state_update equivocator_IM s eqv si', _om') =
(s', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
is_equivocating_state (IM i) (s i)
  inversion Ht.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
eqv: index
leqv: vlabel (equivocator_IM eqv)
s: composite_state equivocator_IM
om: option message
s': composite_state equivocator_IM
om': option message
si': equivocator_state (IM eqv)
_om': option message
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', _om')
Ht: (state_update equivocator_IM s eqv si', _om') =
(s', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s' i)
H0: state_update equivocator_IM s eqv si' = s'
H1: _om' = om'
is_equivocating_state (IM i) (s i) subst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
eqv: index
leqv: vlabel (equivocator_IM eqv)
s: composite_state equivocator_IM
om, om': option message
si': equivocator_state (IM eqv)
Ht: (state_update equivocator_IM s eqv si', om') =
(state_update equivocator_IM s eqv si', om')
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i)
  (state_update equivocator_IM s eqv si' i)
is_equivocating_state (IM i) (s i) clear Ht.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
eqv: index
leqv: vlabel (equivocator_IM eqv)
s: composite_state equivocator_IM
om, om': option message
si': equivocator_state (IM eqv)
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i)
  (state_update equivocator_IM s eqv si' i)
is_equivocating_state (IM i) (s i)
  destruct (decide (i = eqv)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
eqv: index
leqv: vlabel (equivocator_IM eqv)
s: composite_state equivocator_IM
om, om': option message
si': equivocator_state (IM eqv)
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i)
  (state_update equivocator_IM s eqv si' i)
e: i = eqv
is_equivocating_state (IM i) (s i)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
eqv: index
leqv: vlabel (equivocator_IM eqv)
s: composite_state equivocator_IM
om, om': option message
si': equivocator_state (IM eqv)
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i)
  (state_update equivocator_IM s eqv si' i)
n: i ≠ eqv
is_equivocating_state (IM i) (s i)
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
eqv: index
leqv: vlabel (equivocator_IM eqv)
s: composite_state equivocator_IM
om, om': option message
si': equivocator_state (IM eqv)
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i)
  (state_update equivocator_IM s eqv si' i)
e: i = eqv
is_equivocating_state (IM i) (s i) subst.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
eqv: index
leqv: vlabel (equivocator_IM eqv)
s: composite_state equivocator_IM
om, om': option message
si': equivocator_state (IM eqv)
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
Hi: is_equivocating_state (IM eqv)
  (state_update equivocator_IM s eqv si' eqv)
is_equivocating_state (IM eqv) (s eqv) rewrite state_update_eq in Hi.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
eqv: index
leqv: vlabel (equivocator_IM eqv)
s: composite_state equivocator_IM
om, om': option message
si': equivocator_state (IM eqv)
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
Hi: is_equivocating_state (IM eqv) si'
is_equivocating_state (IM eqv) (s eqv)
    apply
      (zero_descriptor_transition_reflects_equivocating_state
        (IM eqv) _ _ _ _ _ Hti _ Hzero Hi).
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
eqv: index
leqv: vlabel (equivocator_IM eqv)
s: composite_state equivocator_IM
om, om': option message
si': equivocator_state (IM eqv)
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i)
  (state_update equivocator_IM s eqv si' i)
n: i ≠ eqv
is_equivocating_state (IM i) (s i) rewrite state_update_neq in Hi by congruence.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
eqv: index
leqv: vlabel (equivocator_IM eqv)
s: composite_state equivocator_IM
om, om': option message
si': equivocator_state (IM eqv)
Hti: equivocator_transition (IM eqv) leqv (s eqv, om) =
(si', om')
Hs: state_has_fixed_equivocation s
li: vlabel (IM (projT1 (existT eqv leqv)))
Hzero: projT2 (existT eqv leqv) = ContinueWith 0 li
i: index
Hi: is_equivocating_state (IM i) (s i)
n: i ≠ eqv
is_equivocating_state (IM i) (s i)
    assumption.
Qed.

If a future state has fixed equivocation, then so must the current state.

Lemma in_futures_reflects_fixed_equivocation
  (s1 s2 : composite_state equivocator_IM)
  (Hfutures : in_futures (pre_loaded_with_all_messages_vlsm (free_composite_vlsm equivocator_IM)) s1 s2)
  : state_has_fixed_equivocation s2 -> state_has_fixed_equivocation s1.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s1, s2: composite_state equivocator_IM
Hfutures: in_futures
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
  s1 s2
state_has_fixed_equivocation s2
→ state_has_fixed_equivocation s1
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s1, s2: composite_state equivocator_IM
Hfutures: in_futures
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
  s1 s2
state_has_fixed_equivocation s2
→ state_has_fixed_equivocation s1
  destruct Hfutures as [tr Htr].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s1, s2: composite_state equivocator_IM
tr: list transition_item
Htr: finite_valid_trace_from_to
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s1 s2
  tr
state_has_fixed_equivocation s2
→ state_has_fixed_equivocation s1
  induction Htr;[trivial|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
iom, oom: option message
l: label
Ht: input_valid_transition
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) l
  (s', iom) (s, oom)
IHHtr: state_has_fixed_equivocation f
→ state_has_fixed_equivocation s
state_has_fixed_equivocation f
→ state_has_fixed_equivocation s'
  intros Hf.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
iom, oom: option message
l: label
Ht: input_valid_transition
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) l
  (s', iom) (s, oom)
IHHtr: state_has_fixed_equivocation f
→ state_has_fixed_equivocation s
Hf: state_has_fixed_equivocation f
state_has_fixed_equivocation s' specialize (IHHtr Hf).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
iom, oom: option message
l: label
Ht: input_valid_transition
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) l
  (s', iom) (s, oom)
IHHtr: state_has_fixed_equivocation s
Hf: state_has_fixed_equivocation f
state_has_fixed_equivocation s' revert IHHtr.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
iom, oom: option message
l: label
Ht: input_valid_transition
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) l
  (s', iom) (s, oom)
Hf: state_has_fixed_equivocation f
state_has_fixed_equivocation s
→ state_has_fixed_equivocation s'
  apply transitivity.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
iom, oom: option message
l: label
Ht: input_valid_transition
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) l
  (s', iom) (s, oom)
Hf: state_has_fixed_equivocation f
equivocating_indices IM index_listing s'
⊆ equivocating_indices IM index_listing s
  destruct Ht as [_ Ht].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
iom, oom: option message
l: label
Ht: transition l (s', iom) = (s, oom)
Hf: state_has_fixed_equivocation f
equivocating_indices IM index_listing s'
⊆ equivocating_indices IM index_listing s
  revert Ht.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM)) s f
  tl
s': state
iom, oom: option message
l: label
Hf: state_has_fixed_equivocation f
transition l (s', iom) = (s, oom)
→ equivocating_indices IM index_listing s'
  ⊆ equivocating_indices IM index_listing s
  apply equivocators_transition_preserves_equivocating_indices.
Qed.

Composition of equivocators with no message equivocation and a fixed set of machines allowed to state-equivocate.

Definition equivocators_fixed_equivocations_vlsm
  : VLSM message
  :=
  composite_vlsm equivocator_IM equivocators_fixed_equivocations_constraint.

Lemma not_equivocating_index_has_singleton_state
  (s : composite_state equivocator_IM)
  (Hs : valid_state_prop equivocators_fixed_equivocations_vlsm s)
  (i : index)
  (Hi : i ∉ equivocating)
  : is_singleton_state (IM i) (s i).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
Hs: valid_state_prop
  equivocators_fixed_equivocations_vlsm s
i: index
Hi: i ∉ equivocating
is_singleton_state (IM i) (s i)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
Hs: valid_state_prop
  equivocators_fixed_equivocations_vlsm s
i: index
Hi: i ∉ equivocating
is_singleton_state (IM i) (s i)
  apply valid_state_has_trace in Hs.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
Hs: ∃ (is : state) (tr : list transition_item),
  finite_valid_trace_init_to
    equivocators_fixed_equivocations_vlsm is s tr
i: index
Hi: i ∉ equivocating
is_singleton_state (IM i) (s i)
  destruct Hs as [is [tr [Htr His]]].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
His: initial_state_prop is
i: index
Hi: i ∉ equivocating
is_singleton_state (IM i) (s i)
  assert (is_singleton_state (IM i) (is i))
    by (apply His).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
His: initial_state_prop is
i: index
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (is i)
is_singleton_state (IM i) (s i)
  clear His.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
i: index
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (is i)
is_singleton_state (IM i) (s i)
  induction Htr.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: state
Hs: valid_state_prop
  equivocators_fixed_equivocations_vlsm s
i: index
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s i)
is_singleton_state (IM i) (s i)
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state
iom, oom: option message
l: label
Ht: input_valid_transition
  equivocators_fixed_equivocations_vlsm l
  (s', iom) (s, oom)
i: index
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
IHHtr: is_singleton_state (IM i) (s i)
→ is_singleton_state (IM i) (f i)
is_singleton_state (IM i) (f i)
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: state
Hs: valid_state_prop
  equivocators_fixed_equivocations_vlsm s
i: index
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s i)
is_singleton_state (IM i) (s i) assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state
iom, oom: option message
l: label
Ht: input_valid_transition
  equivocators_fixed_equivocations_vlsm l
  (s', iom) (s, oom)
i: index
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
IHHtr: is_singleton_state (IM i) (s i)
→ is_singleton_state (IM i) (f i)
is_singleton_state (IM i) (f i) apply IHHtr; clear IHHtr.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state
iom, oom: option message
l: label
Ht: input_valid_transition
  equivocators_fixed_equivocations_vlsm l
  (s', iom) (s, oom)
i: index
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
is_singleton_state (IM i) (s i)
    destruct Ht as [[_ [_ [_ [_ Hfixed]]]] Ht].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state
iom, oom: option message
l: label
Hfixed: state_has_fixed_equivocation
  (composite_transition equivocator_IM l
     (s', iom)).1
Ht: transition l (s', iom) = (s, oom)
i: index
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
is_singleton_state (IM i) (s i)
    simpl in *.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: _composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': _composite_state equivocator_IM
iom, oom: option message
l: _composite_label equivocator_IM
Hfixed: state_has_fixed_equivocation
  (let (i, li) := l in
   let (si', om') :=
     vtransition (equivocator_IM i) li
       (s' i, iom) in
   (state_update equivocator_IM s' i si',
   om')).1
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition (equivocator_IM i) li (s' i, iom) in
 (state_update equivocator_IM s' i si', om')) =
(s, oom)
i: index
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
is_singleton_state (IM i) (s i) rewrite Ht in Hfixed.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: _composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': _composite_state equivocator_IM
iom, oom: option message
l: _composite_label equivocator_IM
Hfixed: state_has_fixed_equivocation (s, oom).1
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition (equivocator_IM i) li (s' i, iom) in
 (state_update equivocator_IM s' i si', om')) =
(s, oom)
i: index
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
is_singleton_state (IM i) (s i) clear Ht.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: _composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': _composite_state equivocator_IM
iom, oom: option message
l: _composite_label equivocator_IM
Hfixed: state_has_fixed_equivocation (s, oom).1
i: index
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
is_singleton_state (IM i) (s i) simpl in Hfixed.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: _composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': _composite_state equivocator_IM
iom, oom: option message
l: _composite_label equivocator_IM
Hfixed: state_has_fixed_equivocation s
i: index
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
is_singleton_state (IM i) (s i)
    specialize (Hfixed i).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: _composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': _composite_state equivocator_IM
iom, oom: option message
l: _composite_label equivocator_IM
i: index
Hfixed: i ∈ equivocating_indices IM index_listing s
→ i ∈ equivocating
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
is_singleton_state (IM i) (s i)
    destruct (decide (equivocator_state_n (s i) = 1)); [assumption|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: _composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': _composite_state equivocator_IM
iom, oom: option message
l: _composite_label equivocator_IM
i: index
Hfixed: i ∈ equivocating_indices IM index_listing s
→ i ∈ equivocating
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
n: equivocator_state_n (s i) ≠ 1
is_singleton_state (IM i) (s i)
    elim Hi.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: _composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': _composite_state equivocator_IM
iom, oom: option message
l: _composite_label equivocator_IM
i: index
Hfixed: i ∈ equivocating_indices IM index_listing s
→ i ∈ equivocating
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
n: equivocator_state_n (s i) ≠ 1
i ∈ equivocating apply Hfixed.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: _composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': _composite_state equivocator_IM
iom, oom: option message
l: _composite_label equivocator_IM
i: index
Hfixed: i ∈ equivocating_indices IM index_listing s
→ i ∈ equivocating
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
n: equivocator_state_n (s i) ≠ 1
i ∈ equivocating_indices IM index_listing s
    apply elem_of_list_filter.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: _composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': _composite_state equivocator_IM
iom, oom: option message
l: _composite_label equivocator_IM
i: index
Hfixed: i ∈ equivocating_indices IM index_listing s
→ i ∈ equivocating
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
n: equivocator_state_n (s i) ≠ 1
is_equivocating_state (IM i) (s i) ∧ i ∈ index_listing
    split; [|apply elem_of_list_In; apply finite_index].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: _composite_state equivocator_IM
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': _composite_state equivocator_IM
iom, oom: option message
l: _composite_label equivocator_IM
i: index
Hfixed: i ∈ equivocating_indices IM index_listing s
→ i ∈ equivocating
Hi: i ∉ equivocating
H: is_singleton_state (IM i) (s' i)
n: equivocator_state_n (s i) ≠ 1
is_equivocating_state (IM i) (s i)
    assumption.
Qed.

Lemma valid_state_has_fixed_equivocation
  (s : composite_state equivocator_IM)
  (Hs : valid_state_prop equivocators_fixed_equivocations_vlsm s)
  : state_has_fixed_equivocation s.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
Hs: valid_state_prop
  equivocators_fixed_equivocations_vlsm s
state_has_fixed_equivocation s
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
Hs: valid_state_prop
  equivocators_fixed_equivocations_vlsm s
state_has_fixed_equivocation s
  apply valid_state_has_trace in Hs.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
Hs: ∃ (is : state) (tr : list transition_item),
  finite_valid_trace_init_to
    equivocators_fixed_equivocations_vlsm is s tr
state_has_fixed_equivocation s
  destruct Hs as [is [tr [Htr Hinit]]].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
state_has_fixed_equivocation s
  assert (state_has_fixed_equivocation is) as His.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
state_has_fixed_equivocation is
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
His: state_has_fixed_equivocation is
state_has_fixed_equivocation s {message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
state_has_fixed_equivocation is
    intros i Hin.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
i: index
Hin: i ∈ equivocating_indices IM index_listing is
i ∈ equivocating
    apply elem_of_list_filter in Hin.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
i: index
Hin: is_equivocating_state (IM i) (is i)
∧ i ∈ index_listing
i ∈ equivocating
    destruct Hin as [His Hin].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
i: index
His: is_equivocating_state (IM i) (is i)
Hin: i ∈ index_listing
i ∈ equivocating
    unfold is_equivocating_state in His.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
i: index
His: ¬ is_singleton_state (IM i) (is i)
Hin: i ∈ index_listing
i ∈ equivocating
    contradict His.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
i: index
Hin: i ∈ index_listing
is_singleton_state (IM i) (is i)
    apply Hinit.
  }message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
Hinit: initial_state_prop is
His: state_has_fixed_equivocation is
state_has_fixed_equivocation s
  clear Hinit.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: composite_state equivocator_IM
is: state
tr: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm is s tr
His: state_has_fixed_equivocation is
state_has_fixed_equivocation s
  induction Htr.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: state
Hs: valid_state_prop
  equivocators_fixed_equivocations_vlsm s
His: state_has_fixed_equivocation s
state_has_fixed_equivocation s
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state
iom, oom: option message
l: label
Ht: input_valid_transition
  equivocators_fixed_equivocations_vlsm l
  (s', iom) (s, oom)
His: state_has_fixed_equivocation s'
IHHtr: state_has_fixed_equivocation s
→ state_has_fixed_equivocation f
state_has_fixed_equivocation f
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s: state
Hs: valid_state_prop
  equivocators_fixed_equivocations_vlsm s
His: state_has_fixed_equivocation s
state_has_fixed_equivocation s assumption.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state
iom, oom: option message
l: label
Ht: input_valid_transition
  equivocators_fixed_equivocations_vlsm l
  (s', iom) (s, oom)
His: state_has_fixed_equivocation s'
IHHtr: state_has_fixed_equivocation s
→ state_has_fixed_equivocation f
state_has_fixed_equivocation f apply IHHtr; clear IHHtr.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state
iom, oom: option message
l: label
Ht: input_valid_transition
  equivocators_fixed_equivocations_vlsm l
  (s', iom) (s, oom)
His: state_has_fixed_equivocation s'
state_has_fixed_equivocation s
    destruct Ht as [[_ [_ [_ [_ Hv]]]] Ht].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state
iom, oom: option message
l: label
Hv: state_has_fixed_equivocation
  (composite_transition equivocator_IM l
     (s', iom)).1
Ht: transition l (s', iom) = (s, oom)
His: state_has_fixed_equivocation s'
state_has_fixed_equivocation s
    setoid_rewrite Ht in Hv.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
s, f: state
tl: list transition_item
Htr: finite_valid_trace_from_to
  equivocators_fixed_equivocations_vlsm s f tl
s': state
iom, oom: option message
l: label
Hv: state_has_fixed_equivocation (s, oom).1
Ht: transition l (s', iom) = (s, oom)
His: state_has_fixed_equivocation s'
state_has_fixed_equivocation s
    assumption.
Qed.

Inclusion in the free composition.

Lemma equivocators_fixed_equivocations_vlsm_incl_free
  : VLSM_incl equivocators_fixed_equivocations_vlsm (free_composite_vlsm equivocator_IM).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
VLSM_incl equivocators_fixed_equivocations_vlsm
  (free_composite_vlsm equivocator_IM)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
VLSM_incl equivocators_fixed_equivocations_vlsm
  (free_composite_vlsm equivocator_IM)
  apply constraint_subsumption_incl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
input_valid_constraint_subsumption equivocator_IM
  equivocators_fixed_equivocations_constraint
  (free_constraint equivocator_IM)
  intro; intros.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label equivocator_IM
som: (composite_state equivocator_IM * option message)%type
H: input_valid
  (composite_vlsm equivocator_IM
     equivocators_fixed_equivocations_constraint)
  l som
free_constraint equivocator_IM l som exact I.
Qed.

Inclusion into the preloaded free composition.

Lemma equivocators_fixed_equivocations_vlsm_incl_PreFree
  : VLSM_incl equivocators_fixed_equivocations_vlsm (pre_loaded_with_all_messages_vlsm (free_composite_vlsm equivocator_IM)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
VLSM_incl equivocators_fixed_equivocations_vlsm
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
VLSM_incl equivocators_fixed_equivocations_vlsm
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
  apply VLSM_incl_trans with (machine (free_composite_vlsm equivocator_IM)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
VLSM_incl
  (mk_vlsm
     (composite_vlsm_machine equivocator_IM
        equivocators_fixed_equivocations_constraint))
  (mk_vlsm
     (machine (free_composite_vlsm equivocator_IM)))
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
VLSM_incl
  (mk_vlsm
     (machine (free_composite_vlsm equivocator_IM)))
  (mk_vlsm
     (pre_loaded_with_all_messages_vlsm_machine
        (free_composite_vlsm equivocator_IM)))
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
VLSM_incl
  (mk_vlsm
     (composite_vlsm_machine equivocator_IM
        equivocators_fixed_equivocations_constraint))
  (mk_vlsm
     (machine (free_composite_vlsm equivocator_IM))) apply equivocators_fixed_equivocations_vlsm_incl_free.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
VLSM_incl
  (mk_vlsm
     (machine (free_composite_vlsm equivocator_IM)))
  (mk_vlsm
     (pre_loaded_with_all_messages_vlsm_machine
        (free_composite_vlsm equivocator_IM))) apply vlsm_incl_pre_loaded_with_all_messages_vlsm.
Qed.

Inclusion of preloaded machine into the preloaded free composition.

Lemma preloaded_equivocators_fixed_equivocations_vlsm_incl_free
  : VLSM_incl (pre_loaded_with_all_messages_vlsm equivocators_fixed_equivocations_vlsm) (pre_loaded_with_all_messages_vlsm (free_composite_vlsm equivocator_IM)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
VLSM_incl
  (pre_loaded_with_all_messages_vlsm
     equivocators_fixed_equivocations_vlsm)
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
VLSM_incl
  (pre_loaded_with_all_messages_vlsm
     equivocators_fixed_equivocations_vlsm)
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm equivocator_IM))
  apply basic_VLSM_incl_preloaded; intro; intros; [assumption| |assumption].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: label
s: state
om: option message
H: vvalid
  (mk_vlsm
     (composite_vlsm_machine equivocator_IM
        equivocators_fixed_equivocations_constraint))
  l (s, om)
vvalid
  (mk_vlsm
     (composite_vlsm_machine equivocator_IM
        (free_constraint equivocator_IM))) (id l)
  (id s, om)
  split; [|exact I].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: label
s: state
om: option message
H: vvalid
  (mk_vlsm
     (composite_vlsm_machine equivocator_IM
        equivocators_fixed_equivocations_constraint))
  l (s, om)
composite_valid equivocator_IM (id l) (id s, om) apply H.
Qed.

Inclusion in the composition of equivocators with no message equivocation (no restriction on state equivocation).

Lemma equivocators_fixed_equivocations_vlsm_incl_no_equivocations
  : VLSM_incl equivocators_fixed_equivocations_vlsm (equivocators_no_equivocations_vlsm IM Hbs).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
VLSM_incl equivocators_fixed_equivocations_vlsm
  (equivocators_no_equivocations_vlsm IM Hbs)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
VLSM_incl equivocators_fixed_equivocations_vlsm
  (equivocators_no_equivocations_vlsm IM Hbs)
  apply constraint_subsumption_incl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
input_valid_constraint_subsumption
  (Common.equivocator_IM IM)
  equivocators_fixed_equivocations_constraint
  (equivocators_no_equivocations_constraint IM Hbs)
  intros l (s,om) Hv.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
l: composite_label (Common.equivocator_IM IM)
s: composite_state (Common.equivocator_IM IM)
om: option message
Hv: input_valid
  (composite_vlsm (Common.equivocator_IM IM)
     equivocators_fixed_equivocations_constraint)
  l (s, om)
equivocators_no_equivocations_constraint IM Hbs l
  (s, om) apply Hv.
Qed.

End equivocators_fixed_equivocations_vlsm.

Section fixed_equivocation_with_fullnode.

This section instantiates the full_node_condition_for_admissible_equivocators by choosing the admissible indices to be the fixed set of nodes allowed to equivocate, and then shows that this constraint is stronger than the fixed_equivocation_constraint.

Context setting the stage for, and instantiating the full_node_condition_for_admissible_equivocators. It requires that the nodes have has_been_sent and has_been_received capabilities, that the number of nodes is finite

  Context
    {message : Type}
    {index : Type}
    {IndEqDec : EqDecision index}
    (IM : index -> VLSM message)
    (Hbs : forall i : index, HasBeenSentCapability (IM i))
    (Hbr : forall i : index, HasBeenReceivedCapability (IM i))
    (Hbo : forall i : index, HasBeenObservedCapability (IM i) := fun i : index => HasBeenObservedCapability_from_sent_received (IM i))
    (equivocator_IM := equivocator_IM IM)
    (index_listing : list index)
    (finite_index : Listing index_listing)
    (equivocating : set index)
    (admissible_index := fun s i => i ∈ equivocating)
    (full_node_constraint
      := full_node_condition_for_admissible_equivocators IM Hbs Hbr admissible_index)
    (full_node_constraint_alt
      := full_node_condition_for_admissible_equivocators_alt IM Hbs Hbr admissible_index)
    .

Context for the fixed_equivocation_constraint. Additionally to the above it requires that equality on messages is decidable and that the set of VLSMs allowed to equivocate is non-empty.

  Context
    `{EqDecision message}
    (equivocating_index : Type := sub_index equivocating)
    (equivocating_IM := sub_IM IM equivocating)
    (fixed_equivocation_constraint : composite_label IM  -> composite_state IM * option message -> Prop
      := fixed_equivocation_constraint IM Hbs Hbr equivocating)
    (Free := free_composite_vlsm IM)
    (Free_HasBeenSentCapability : HasBeenSentCapability Free := free_composite_HasBeenSentCapability IM finite_index Hbs)
    (Free_HasBeenReceivedCapability : HasBeenReceivedCapability Free := free_composite_HasBeenReceivedCapability IM finite_index Hbr)
    (Free_HasBeenObservedCapability : HasBeenObservedCapability Free := free_composite_HasBeenObservedCapability IM finite_index Hbo)
    .

  Existing Instance Free_HasBeenObservedCapability.

The full_node_constraint_alt is stronger than the fixed_equivocation_constraint.

  Lemma fixed_equivocation_constraint_subsumption_alt
    : strong_constraint_subsumption IM full_node_constraint_alt fixed_equivocation_constraint.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
strong_constraint_subsumption IM
  full_node_constraint_alt
  fixed_equivocation_constraint
  Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
strong_constraint_subsumption IM
  full_node_constraint_alt
  fixed_equivocation_constraint
    intros l (s, [m|]) Hc ; [| exact I].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
Hc: full_node_constraint_alt l (s, Some m)
fixed_equivocation_constraint l (s, Some m)
    destruct Hc as [Hno_equiv | [i [Hi Hm]]]; [left; assumption|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
i: index
Hi: admissible_index s i
Hm: node_generated_without_further_equivocation_alt
  IM Hbs Hbr s m i
fixed_equivocation_constraint l (s, Some m)
    unfold node_generated_without_further_equivocation_alt in Hm.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
i: index
Hi: admissible_index s i
Hm: can_emit
  (pre_loaded_vlsm (IM i)
     (composite_has_been_observed IM
        (λ i : index,
           HasBeenObservedCapability_from_sent_received
             (IM i)) s)) m
fixed_equivocation_constraint l (s, Some m)
    right.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
i: index
Hi: admissible_index s i
Hm: can_emit
  (pre_loaded_vlsm (IM i)
     (composite_has_been_observed IM
        (λ i : index,
           HasBeenObservedCapability_from_sent_received
             (IM i)) s)) m
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating s) m
    remember (composite_has_been_observed IM _ _) as P.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
i: index
Hi: admissible_index s i
P: message → Prop
HeqP: P =
composite_has_been_observed IM
  (λ i : index,
     HasBeenObservedCapability_from_sent_received
       (IM i)) s
Hm: can_emit (pre_loaded_vlsm (IM i) P) m
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating s) m
    specialize (can_emit_composite_free_lift equivocating_IM P P) as Hemit.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
i: index
Hi: admissible_index s i
P: message → Prop
HeqP: P =
composite_has_been_observed IM
  (λ i : index,
     HasBeenObservedCapability_from_sent_received
       (IM i)) s
Hm: can_emit (pre_loaded_vlsm (IM i) P) m
Hemit: (∀ m : message, P m → P m)
→ ∀ (j : sub_index equivocating) (m : message),
    can_emit
      (pre_loaded_vlsm (equivocating_IM j) P)
      m
    → can_emit
        (pre_loaded_vlsm
           (free_composite_vlsm
              equivocating_IM) P) m
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating s) m
    spec Hemit.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
i: index
Hi: admissible_index s i
P: message → Prop
HeqP: P =
composite_has_been_observed IM
  (λ i : index,
     HasBeenObservedCapability_from_sent_received
       (IM i)) s
Hm: can_emit (pre_loaded_vlsm (IM i) P) m
Hemit: (∀ m : message, P m → P m)
→ ∀ (j : sub_index equivocating) (m : message),
    can_emit
      (pre_loaded_vlsm (equivocating_IM j) P)
      m
    → can_emit
        (pre_loaded_vlsm
           (free_composite_vlsm
              equivocating_IM) P) m
∀ m : message, P m → P m
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
i: index
Hi: admissible_index s i
P: message → Prop
HeqP: P =
composite_has_been_observed IM
  (λ i : index,
     HasBeenObservedCapability_from_sent_received
       (IM i)) s
Hm: can_emit (pre_loaded_vlsm (IM i) P) m
Hemit: ∀ (j : sub_index equivocating) (m : message),
  can_emit
    (pre_loaded_vlsm (equivocating_IM j) P) m
  → can_emit
      (pre_loaded_vlsm
         (free_composite_vlsm equivocating_IM)
         P) m
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating s) m {message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
i: index
Hi: admissible_index s i
P: message → Prop
HeqP: P =
composite_has_been_observed IM
  (λ i : index,
     HasBeenObservedCapability_from_sent_received
       (IM i)) s
Hm: can_emit (pre_loaded_vlsm (IM i) P) m
Hemit: (∀ m : message, P m → P m)
→ ∀ (j : sub_index equivocating) (m : message),
    can_emit
      (pre_loaded_vlsm (equivocating_IM j) P)
      m
    → can_emit
        (pre_loaded_vlsm
           (free_composite_vlsm
              equivocating_IM) P) m
∀ m : message, P m → P m intro.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
i: index
Hi: admissible_index s i
P: message → Prop
HeqP: P =
composite_has_been_observed IM
  (λ i : index,
     HasBeenObservedCapability_from_sent_received
       (IM i)) s
Hm: can_emit (pre_loaded_vlsm (IM i) P) m
Hemit: (∀ m : message, P m → P m)
→ ∀ (j : sub_index equivocating) (m : message),
    can_emit
      (pre_loaded_vlsm (equivocating_IM j) P)
      m
    → can_emit
        (pre_loaded_vlsm
           (free_composite_vlsm
              equivocating_IM) P) m
m0: message
P m0 → P m0 apply id. }message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
i: index
Hi: admissible_index s i
P: message → Prop
HeqP: P =
composite_has_been_observed IM
  (λ i : index,
     HasBeenObservedCapability_from_sent_received
       (IM i)) s
Hm: can_emit (pre_loaded_vlsm (IM i) P) m
Hemit: ∀ (j : sub_index equivocating) (m : message),
  can_emit
    (pre_loaded_vlsm (equivocating_IM j) P) m
  → can_emit
      (pre_loaded_vlsm
         (free_composite_vlsm equivocating_IM)
         P) m
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating s) m
    specialize (Hemit (dec_exist _  i Hi) m).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
i: index
Hi: admissible_index s i
P: message → Prop
HeqP: P =
composite_has_been_observed IM
  (λ i : index,
     HasBeenObservedCapability_from_sent_received
       (IM i)) s
Hm: can_emit (pre_loaded_vlsm (IM i) P) m
Hemit: can_emit
  (pre_loaded_vlsm
     (equivocating_IM
        (dec_exist (admissible_index s) i Hi))
     P) m
→ can_emit
    (pre_loaded_vlsm
       (free_composite_vlsm equivocating_IM) P)
    m
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating s) m subst P.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
i: index
Hi: admissible_index s i
Hemit: can_emit
  (pre_loaded_vlsm
     (equivocating_IM
        (dec_exist (admissible_index s) i Hi))
     (composite_has_been_observed IM
        (λ i : index,
           HasBeenObservedCapability_from_sent_received
             (IM i)) s)) m
→ can_emit
    (pre_loaded_vlsm
       (free_composite_vlsm equivocating_IM)
       (composite_has_been_observed IM
          (λ i : index,
             HasBeenObservedCapability_from_sent_received
               (IM i)) s)) m
Hm: can_emit
  (pre_loaded_vlsm (IM i)
     (composite_has_been_observed IM
        (λ i : index,
           HasBeenObservedCapability_from_sent_received
             (IM i)) s)) m
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating s) m
    apply Hemit.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
l: composite_label IM
s: composite_state IM
m: message
i: index
Hi: admissible_index s i
Hemit: can_emit
  (pre_loaded_vlsm
     (equivocating_IM
        (dec_exist (admissible_index s) i Hi))
     (composite_has_been_observed IM
        (λ i : index,
           HasBeenObservedCapability_from_sent_received
             (IM i)) s)) m
→ can_emit
    (pre_loaded_vlsm
       (free_composite_vlsm equivocating_IM)
       (composite_has_been_observed IM
          (λ i : index,
             HasBeenObservedCapability_from_sent_received
               (IM i)) s)) m
Hm: can_emit
  (pre_loaded_vlsm (IM i)
     (composite_has_been_observed IM
        (λ i : index,
           HasBeenObservedCapability_from_sent_received
             (IM i)) s)) m
can_emit
  (pre_loaded_vlsm
     (equivocating_IM
        (dec_exist (admissible_index s) i Hi))
     (composite_has_been_observed IM
        (λ i : index,
           HasBeenObservedCapability_from_sent_received
             (IM i)) s)) m
    assumption.
  Qed.

If all nodes have the cannot_resend_message_stepwise_property, then the full node constraint is stronger than the fixed_equivocation_constraint.

(by reduction to the result above)

  Lemma fixed_equivocation_constraint_subsumption
    (Hno_resend : forall i : index, cannot_resend_message_stepwise_prop (IM i))
    : preloaded_constraint_subsumption IM full_node_constraint fixed_equivocation_constraint.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Hno_resend: ∀ i : index,
  cannot_resend_message_stepwise_prop
    (IM i)
preloaded_constraint_subsumption IM
  full_node_constraint fixed_equivocation_constraint
  Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Hno_resend: ∀ i : index,
  cannot_resend_message_stepwise_prop
    (IM i)
preloaded_constraint_subsumption IM
  full_node_constraint fixed_equivocation_constraint
    intros l (s, om) Hv.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Hno_resend: ∀ i : index,
  cannot_resend_message_stepwise_prop
    (IM i)
l: composite_label IM
s: state
om: option message
Hv: input_valid
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm IM full_node_constraint)) l
  (s, om)
fixed_equivocation_constraint l (s, om)
    apply fixed_equivocation_constraint_subsumption_alt.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Hno_resend: ∀ i : index,
  cannot_resend_message_stepwise_prop
    (IM i)
l: composite_label IM
s: state
om: option message
Hv: input_valid
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm IM full_node_constraint)) l
  (s, om)
full_node_constraint_alt l (s, om)
    apply
      (full_node_condition_for_admissible_equivocators_subsumption
        IM Hbs Hbr admissible_index Hno_resend).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocator_IM:= Common.equivocator_IM IM: index → VLSM message
index_listing: list index
finite_index: Listing index_listing
equivocating: set index
admissible_index:= λ (_ : composite_state IM) 
  (i : index), i ∈ equivocating: composite_state IM → index → Prop
full_node_constraint:= full_node_condition_for_admissible_equivocators
  IM Hbs Hbr admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
full_node_constraint_alt:= full_node_condition_for_admissible_equivocators_alt
  IM Hbs Hbr
  admissible_index: composite_label IM
→ composite_state IM *
  option message → Prop
EqDecision0: EqDecision message
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
fixed_equivocation_constraint:= FixedSetEquivocation.fixed_equivocation_constraint
  IM Hbs Hbr
  equivocating: composite_label IM
→ composite_state IM *
  option message
  → Prop
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Hno_resend: ∀ i : index,
  cannot_resend_message_stepwise_prop
    (IM i)
l: composite_label IM
s: state
om: option message
Hv: input_valid
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm IM full_node_constraint)) l
  (s, om)
input_valid
  (pre_loaded_with_all_messages_vlsm
     (composite_vlsm IM
        (full_node_condition_for_admissible_equivocators
           IM Hbs Hbr admissible_index))) l (s, om)
    assumption.
  Qed.

End fixed_equivocation_with_fullnode.

Section from_equivocators_to_nodes.

From composition of equivocators to composition of simple nodes

In this section we show that the projection of valid_traces of a composition of equivocators with a fixed equivocation constraint are valid_traces for the composition of nodes with a similar fixed equivocation constraint.

Context
  {message : Type}
  `{EqDecision message}
  {index : Type}
  {IndEqDec : EqDecision index}
  (IM : index -> VLSM message)
  (Hbs : forall i : index, HasBeenSentCapability (IM i))
  (Hbr : forall i : index, HasBeenReceivedCapability (IM i))
  (Hbo : forall i : index, HasBeenObservedCapability (IM i)
    := fun i => HasBeenObservedCapability_from_sent_received (IM i))
  (equivocating : set index)
  {index_listing : list index}
  (finite_index : Listing index_listing)
  (XE : VLSM message := equivocators_fixed_equivocations_vlsm IM Hbs index_listing equivocating)
  (X : VLSM message := fixed_equivocation_vlsm_composition IM Hbs Hbr equivocating)
  (equivocators_free_Hbs := free_composite_HasBeenSentCapability (equivocator_IM IM) finite_index (equivocator_Hbs IM Hbs))
  (FreeE : VLSM message := free_composite_vlsm (equivocator_IM IM))
  (FreeE_HasBeenSentCapability : HasBeenSentCapability FreeE := free_composite_HasBeenSentCapability (equivocator_IM IM) finite_index (equivocator_Hbs IM Hbs))
  (Hdec_init : forall i, vdecidable_initial_messages_prop (IM i))
  (Free := free_composite_vlsm IM)
  (Free_HasBeenSentCapability : HasBeenSentCapability Free := free_composite_HasBeenSentCapability IM finite_index Hbs)
  (Free_HasBeenReceivedCapability : HasBeenReceivedCapability Free := free_composite_HasBeenReceivedCapability IM finite_index Hbr)
  (Free_HasBeenObservedCapability : HasBeenObservedCapability Free := free_composite_HasBeenObservedCapability IM finite_index Hbo)
  (Free_no_additional_equivocation_decidable := no_additional_equivocations_dec Free)
  (index_equivocating_prop : index -> Prop := sub_index_prop equivocating)
  (equivocating_index : Type := sub_index equivocating)
  (equivocating_IM := sub_IM IM equivocating)
  (free_equivocating_vlsm_composition : VLSM message := free_composite_vlsm equivocating_IM)
  (sub_equivocator_IM := sub_IM (equivocator_IM IM) equivocating)
  .

proper_equivocator_descriptors strengthen for fixed-equivocation. We require that if the index is not equivocating, than the corresponding descriptor is a zero_descriptor.

Definition proper_fixed_equivocator_descriptors
  (eqv_descriptors : equivocator_descriptors IM)
  (s : state)
  : Prop
  := proper_equivocator_descriptors IM eqv_descriptors s /\
    forall i, i ∉ equivocating -> eqv_descriptors i = Existing 0.

not_equivocating_equivocator_descriptors satisfy the proper_fixed_equivocator_descriptors property.

Lemma not_equivocating_equivocator_descriptors_proper_fixed
  (s : composite_state (equivocator_IM IM))
  (Hs : valid_state_prop XE s)
  (eqv_descriptors : equivocator_descriptors IM)
  (Heqv_descriptors : not_equivocating_equivocator_descriptors IM eqv_descriptors s)
  : proper_fixed_equivocator_descriptors eqv_descriptors s.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors s
proper_fixed_equivocator_descriptors eqv_descriptors s
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors s
proper_fixed_equivocator_descriptors eqv_descriptors s
  apply not_equivocating_equivocator_descriptors_proper in Heqv_descriptors as Hproper.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors s
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
proper_fixed_equivocator_descriptors eqv_descriptors s
  split; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors s
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
∀ i : index,
  i ∉ equivocating → eqv_descriptors i = Existing 0
  intros i Hi.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors s
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
i: index
Hi: i ∉ equivocating
eqv_descriptors i = Existing 0
  specialize (not_equivocating_index_has_singleton_state IM Hbs _ finite_index equivocating _ Hs _ Hi)
    as Hzero.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors s
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
i: index
Hi: i ∉ equivocating
Hzero: is_singleton_state (IM i) (s i)
eqv_descriptors i = Existing 0
  unfold is_singleton_state in Hzero.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
Heqv_descriptors: not_equivocating_equivocator_descriptors
  IM eqv_descriptors s
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
i: index
Hi: i ∉ equivocating
Hzero: equivocator_state_n (s i) = 1
eqv_descriptors i = Existing 0
  specialize (Heqv_descriptors i).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
i: index
Heqv_descriptors: existing_descriptor 
  (IM i) (eqv_descriptors i) 
  (s i)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
Hi: i ∉ equivocating
Hzero: equivocator_state_n (s i) = 1
eqv_descriptors i = Existing 0 unfold existing_descriptor in Heqv_descriptors.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
i: index
Heqv_descriptors: match eqv_descriptors i with
| NewMachine _ => False
| Existing i0 =>
    is_Some
      (equivocator_state_project
         (s i) i0)
end
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
Hi: i ∉ equivocating
Hzero: equivocator_state_n (s i) = 1
eqv_descriptors i = Existing 0
  destruct (eqv_descriptors i); [contradiction|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
i: index
n: nat
Heqv_descriptors: is_Some
  (equivocator_state_project 
     (s i) n)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
Hi: i ∉ equivocating
Hzero: equivocator_state_n (s i) = 1
Existing n = Existing 0
  destruct Heqv_descriptors as [s_i_n Heqv_descriptors].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
i: index
n: nat
s_i_n: vstate (IM i)
Heqv_descriptors: equivocator_state_project (s i) n =
Some s_i_n
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
Hi: i ∉ equivocating
Hzero: equivocator_state_n (s i) = 1
Existing n = Existing 0
  apply equivocator_state_project_Some_rev in Heqv_descriptors.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
eqv_descriptors: equivocator_descriptors IM
i: index
n: nat
s_i_n: vstate (IM i)
Heqv_descriptors: n < equivocator_state_n (s i)
Hproper: proper_equivocator_descriptors IM
  eqv_descriptors s
Hi: i ∉ equivocating
Hzero: equivocator_state_n (s i) = 1
Existing n = Existing 0
  f_equal; lia.
Qed.

Projections of (valid) traces of the composition of equivocators preserve proper_fixed_equivocator_descriptors.

Lemma equivocators_trace_project_preserves_proper_fixed_equivocator_descriptors
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (Htr : finite_valid_trace_from (pre_loaded_with_all_messages_vlsm FreeE) is tr)
  (s := finite_trace_last is tr)
  (descriptors : equivocator_descriptors IM)
  (idescriptors : equivocator_descriptors IM)
  (trX : list (composite_transition_item IM))
  (HtrX : equivocators_trace_project IM descriptors tr = Some (trX, idescriptors))
  : proper_fixed_equivocator_descriptors descriptors s -> proper_fixed_equivocator_descriptors idescriptors is.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) is
  tr
s:= finite_trace_last is tr: state
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
proper_fixed_equivocator_descriptors descriptors s
→ proper_fixed_equivocator_descriptors idescriptors is
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) is
  tr
s:= finite_trace_last is tr: state
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
proper_fixed_equivocator_descriptors descriptors s
→ proper_fixed_equivocator_descriptors idescriptors is
  intros [Hproper Hfixed].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) is
  tr
s:= finite_trace_last is tr: state
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Hfixed: ∀ i : index,
  i ∉ equivocating
  → descriptors i = Existing 0
proper_fixed_equivocator_descriptors idescriptors is
  specialize
    (preloaded_equivocators_valid_trace_project_proper_initial IM
      _ _ Htr _ _ _ HtrX Hproper
    )
    as Hiproper.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) is
  tr
s:= finite_trace_last is tr: state
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Hfixed: ∀ i : index,
  i ∉ equivocating
  → descriptors i = Existing 0
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
proper_fixed_equivocator_descriptors idescriptors is
  split; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) is
  tr
s:= finite_trace_last is tr: state
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Hfixed: ∀ i : index,
  i ∉ equivocating
  → descriptors i = Existing 0
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
∀ i : index,
  i ∉ equivocating → idescriptors i = Existing 0
  intros i Hi.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) is
  tr
s:= finite_trace_last is tr: state
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Hfixed: ∀ i : index,
  i ∉ equivocating
  → descriptors i = Existing 0
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
i: index
Hi: i ∉ equivocating
idescriptors i = Existing 0 specialize (Hfixed i Hi).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) is
  tr
s:= finite_trace_last is tr: state
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
i: index
Hfixed: descriptors i = Existing 0
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
Hi: i ∉ equivocating
idescriptors i = Existing 0
  revert Hfixed.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) is
  tr
s:= finite_trace_last is tr: state
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
i: index
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
Hi: i ∉ equivocating
descriptors i = Existing 0
→ idescriptors i = Existing 0 clear Hi.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) is
  tr
s:= finite_trace_last is tr: state
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
i: index
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
descriptors i = Existing 0
→ idescriptors i = Existing 0 revert i.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) is
  tr
s:= finite_trace_last is tr: state
descriptors, idescriptors: equivocator_descriptors IM
trX: list (composite_transition_item IM)
HtrX: equivocators_trace_project IM descriptors tr =
Some (trX, idescriptors)
Hproper: proper_equivocator_descriptors IM
  descriptors s
Hiproper: proper_equivocator_descriptors IM
  idescriptors is
∀ i : index,
  descriptors i = Existing 0
  → idescriptors i = Existing 0
  apply (equivocators_trace_project_preserves_zero_descriptors IM) with is tr trX
  ; assumption.
Qed.

Existing Instance equivocators_free_Hbs.

Lemma fixed_equivocators_initial_state_project
  (es : vstate XE)
  (Hes : vinitial_state_prop XE es)
  (eqv_descriptors : equivocator_descriptors IM)
  (Heqv : proper_equivocator_descriptors IM eqv_descriptors es)
  : vinitial_state_prop X (equivocators_state_project IM eqv_descriptors es).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
es: vstate XE
Hes: vinitial_state_prop XE es
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_equivocator_descriptors IM
  eqv_descriptors es
vinitial_state_prop X
  (equivocators_state_project IM eqv_descriptors es)
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
es: vstate XE
Hes: vinitial_state_prop XE es
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_equivocator_descriptors IM
  eqv_descriptors es
vinitial_state_prop X
  (equivocators_state_project IM eqv_descriptors es)
  intro eqv.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
es: vstate XE
Hes: vinitial_state_prop XE es
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_equivocator_descriptors IM
  eqv_descriptors es
eqv: index
vinitial_state_prop (IM eqv)
  (equivocators_state_project IM eqv_descriptors es
     eqv) specialize (Hes eqv).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
es: vstate XE
eqv: index
Hes: vinitial_state_prop 
  (equivocator_IM IM eqv) 
  (es eqv)
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_equivocator_descriptors IM
  eqv_descriptors es
vinitial_state_prop (IM eqv)
  (equivocators_state_project IM eqv_descriptors es
     eqv)
  unfold equivocator_IM in Hes.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
es: vstate XE
eqv: index
Hes: vinitial_state_prop 
  (equivocator_vlsm (IM eqv)) 
  (es eqv)
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_equivocator_descriptors IM
  eqv_descriptors es
vinitial_state_prop (IM eqv)
  (equivocators_state_project IM eqv_descriptors es
     eqv)
  unfold equivocators_state_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
es: vstate XE
eqv: index
Hes: vinitial_state_prop 
  (equivocator_vlsm (IM eqv)) 
  (es eqv)
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_equivocator_descriptors IM
  eqv_descriptors es
vinitial_state_prop (IM eqv)
  (equivocator_state_descriptor_project (es eqv)
     (eqv_descriptors eqv))
  specialize (Heqv eqv).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
es: vstate XE
eqv: index
Hes: vinitial_state_prop 
  (equivocator_vlsm (IM eqv)) 
  (es eqv)
eqv_descriptors: equivocator_descriptors IM
Heqv: proper_descriptor 
  (IM eqv) 
  (eqv_descriptors eqv) 
  (es eqv)
vinitial_state_prop (IM eqv)
  (equivocator_state_descriptor_project (es eqv)
     (eqv_descriptors eqv))
  destruct (eqv_descriptors eqv) as [sn | i]; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
es: vstate XE
eqv: index
Hes: vinitial_state_prop 
  (equivocator_vlsm (IM eqv)) 
  (es eqv)
eqv_descriptors: equivocator_descriptors IM
i: nat
Heqv: proper_descriptor 
  (IM eqv) 
  (Existing i) 
  (es eqv)
vinitial_state_prop (IM eqv)
  (equivocator_state_descriptor_project (es eqv)
     (Existing i))
  destruct Heqv as [esi Heqsi].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
es: vstate XE
eqv: index
Hes: vinitial_state_prop 
  (equivocator_vlsm (IM eqv)) 
  (es eqv)
eqv_descriptors: equivocator_descriptors IM
i: nat
esi: vstate (IM eqv)
Heqsi: equivocator_state_project (es eqv) i =
Some esi
vinitial_state_prop (IM eqv)
  (equivocator_state_descriptor_project (es eqv)
     (Existing i))
  simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
es: vstate XE
eqv: index
Hes: vinitial_state_prop 
  (equivocator_vlsm (IM eqv)) 
  (es eqv)
eqv_descriptors: equivocator_descriptors IM
i: nat
esi: vstate (IM eqv)
Heqsi: equivocator_state_project (es eqv) i =
Some esi
vinitial_state_prop (IM eqv)
  (default (equivocator_state_zero (es eqv))
     (equivocator_state_project (es eqv) i)) rewrite Heqsi.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
es: vstate XE
eqv: index
Hes: vinitial_state_prop 
  (equivocator_vlsm (IM eqv)) 
  (es eqv)
eqv_descriptors: equivocator_descriptors IM
i: nat
esi: vstate (IM eqv)
Heqsi: equivocator_state_project (es eqv) i =
Some esi
vinitial_state_prop (IM eqv)
  (default (equivocator_state_zero (es eqv))
     (Some esi))
  apply equivocator_vlsm_initial_state_preservation_rev with (es eqv) i; assumption.
Qed.

Existing Instance Free_HasBeenObservedCapability.
Existing Instance Free_HasBeenSentCapability.

This is a property of the fixed_equivocation_constraint which also trivially holds for the free constraint. This property is sufficient for proving the _equivocators_valid_trace_project lemma, which lets that lemma be used for both the composition of equivocators with fixed state-equivocation and the free composition.

It basically says that if a message has_been_sent for a state of the composition of equivocators with no-message equivocations and fixed state-equivocations, then any of its projections should be allowed to receive it.

Definition constraint_has_been_sent_prop
  (constraint : composite_label IM -> composite_state IM * option message -> Prop)
  : Prop :=
  forall
    (s : composite_state (equivocator_IM IM))
    (Hs : valid_state_prop XE s)
    (descriptors : equivocator_descriptors IM)
    (Hdescriptors : proper_fixed_equivocator_descriptors descriptors s)
    (sX := @equivocators_state_project _ _ IndEqDec IM descriptors s)
    (m: message)
    (Hm : has_been_sent FreeE s m)
    l,
  constraint l (sX, Some m).

Generic proof that the projection of a trace of the composition of equivocators with no message equivocation and fixed state equivocation is valid w.r.t. the composition of the regular nodes constrained by any constraint satisfying several properties, including the constraint_has_been_sent_property.

The proof proceeds by well founded induction on the length of the trace, performing an analysis on the final transition item of the trace.

It uses the fact that the trace hase no message equivocation to extract a subtrace producing the message being received at the last transition and proves that it's a valid message for the destination machine by using the induction hypothesis (that is why well-founded induction is used rather than a simpler induction principle).

The constraint satisfaction for the projection of the final transition is for now assumes as hypothesis Hconstraint_hbs.

Lemma _equivocators_valid_trace_project
  (final_descriptors : equivocator_descriptors IM)
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (final_state := finite_trace_last is tr)
  (Hproper : proper_fixed_equivocator_descriptors final_descriptors final_state)
  (Htr : finite_valid_trace XE is tr)
  (constraint : composite_label IM -> composite_state IM * option message -> Prop)
  (HconstraintNone : forall l s, constraint l (s, None))
  (Hconstraint_hbs :  constraint_has_been_sent_prop constraint)
  (X' := composite_vlsm IM constraint)
  : exists
    (trX : list (composite_transition_item IM))
    (initial_descriptors : equivocator_descriptors IM)
    (isX := equivocators_state_project IM initial_descriptors is)
    (final_stateX := finite_trace_last isX trX),
    proper_fixed_equivocator_descriptors initial_descriptors is /\
    equivocators_trace_project IM final_descriptors tr = Some (trX, initial_descriptors) /\
    equivocators_state_project IM final_descriptors final_state = final_stateX /\
    finite_valid_trace X' isX trX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
  remember (length tr) as len_tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
len_tr: nat
Heqlen_tr: len_tr = length tr
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
  generalize dependent final_descriptors.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
len_tr: nat
Heqlen_tr: len_tr = length tr
∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors final_state
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : equivocator_descriptors
                               IM),
      let isX :=
        equivocators_state_project IM
          initial_descriptors is in
      let final_stateX := finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
          final_descriptors tr =
        Some (trX, initial_descriptors)
        ∧ equivocators_state_project IM
            final_descriptors final_state =
          final_stateX ∧ finite_valid_trace X' isX trX generalize dependent tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
len_tr: nat
∀ tr : list
         (composite_transition_item
            (equivocator_IM IM)),
  let final_state := finite_trace_last is tr in
  finite_valid_trace XE is tr
  → len_tr = length tr
    → ∀ final_descriptors : equivocator_descriptors IM,
        proper_fixed_equivocator_descriptors
          final_descriptors final_state
        → ∃ (trX : list (composite_transition_item IM)) 
            (initial_descriptors : equivocator_descriptors
                                     IM),
            let isX :=
              equivocators_state_project IM
                initial_descriptors is in
            let final_stateX :=
              finite_trace_last isX trX in
            proper_fixed_equivocator_descriptors
              initial_descriptors is
            ∧ equivocators_trace_project IM
                final_descriptors tr =
              Some (trX, initial_descriptors)
              ∧ equivocators_state_project IM
                  final_descriptors final_state =
                final_stateX
                ∧ finite_valid_trace X' isX trX
  induction len_tr using (well_founded_induction Wf_nat.lt_wf); intros.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
len_tr: nat
H: ∀ y : nat,
  y < len_tr
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
Heqlen_tr: len_tr = length tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
  subst len_tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
H: ∀ y : nat,
  y < length tr
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
  destruct_list_last tr tr' lst Htr_lst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
H: ∀ y : nat,
  y < length []
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = []
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors [] =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is (tr' ++ [lst])
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = 
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
H: ∀ y : nat,
  y < length []
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = []
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors [] =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX clear H.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = []
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors [] =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX subst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors [] =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX unfold final_state in *.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors [] =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        (finite_trace_last is []) =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX exists [], final_descriptors.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
proper_fixed_equivocator_descriptors final_descriptors
  is
∧ equivocators_trace_project IM final_descriptors [] =
  Some ([], final_descriptors)
  ∧ equivocators_state_project IM final_descriptors
      (finite_trace_last is []) =
    finite_trace_last
      (equivocators_state_project IM final_descriptors
         is) []
    ∧ finite_valid_trace X'
        (equivocators_state_project IM
           final_descriptors is) []
    split; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
equivocators_trace_project IM final_descriptors [] =
Some ([], final_descriptors)
∧ equivocators_state_project IM final_descriptors
    (finite_trace_last is []) =
  finite_trace_last
    (equivocators_state_project IM final_descriptors
       is) []
  ∧ finite_valid_trace X'
      (equivocators_state_project IM final_descriptors
         is) [] split; [reflexivity|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
equivocators_state_project IM final_descriptors
  (finite_trace_last is []) =
finite_trace_last
  (equivocators_state_project IM final_descriptors is)
  []
∧ finite_valid_trace X'
    (equivocators_state_project IM final_descriptors
       is) [] split; [reflexivity|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
finite_valid_trace X'
  (equivocators_state_project IM final_descriptors is)
  []
    remember (equivocators_state_project IM final_descriptors is) as isx.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
isx: vstate (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
finite_valid_trace X' isx []
    cut (vinitial_state_prop X' isx).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
isx: vstate (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
vinitial_state_prop X' isx
→ finite_valid_trace X' isx []
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
isx: vstate (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
vinitial_state_prop X' isx
    {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
isx: vstate (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
vinitial_state_prop X' isx
→ finite_valid_trace X' isx [] intro.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
isx: vstate (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
H: vinitial_state_prop X' isx
finite_valid_trace X' isx [] split; [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
isx: vstate (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
H: vinitial_state_prop X' isx
finite_valid_trace_from X' isx [] constructor.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
isx: vstate (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
H: vinitial_state_prop X' isx
valid_state_prop X' isx
      apply valid_state_prop_iff.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
isx: vstate (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
H: vinitial_state_prop X' isx
(∃ is : initial_state, isx = `is)
∨ (∃ (l : label) (som : state * option message) (om' : 
                                                 option
                                                 message),
     input_valid_transition X' l som (isx, om')) left.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
isx: vstate (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
H: vinitial_state_prop X' isx
∃ is : initial_state, isx = `is
      exists (exist _ _ H).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
isx: vstate (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
H: vinitial_state_prop X' isx
isx = `(isx ↾ H) reflexivity.
    }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
isx: vstate (free_composite_vlsm IM)
Heqisx: isx =
equivocators_state_project IM
  final_descriptors is
vinitial_state_prop X' isx
    subst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
vinitial_state_prop X'
  (equivocators_state_project IM final_descriptors is)
    apply fixed_equivocators_initial_state_project; [|apply Hproper].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
final_state:= finite_trace_last is []: state
Htr: finite_valid_trace XE is []
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is [])
vinitial_state_prop XE is
    apply Htr.
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is (tr' ++ [lst])
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX specialize (H (length tr')) as H'.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is (tr' ++ [lst])
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
H': length tr' < length (tr' ++ [lst])
→ ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
    let final_state := finite_trace_last is tr in
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    spec H'.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is (tr' ++ [lst])
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
H': length tr' < length (tr' ++ [lst])
→ ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
    let final_state := finite_trace_last is tr in
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
length tr' < length (tr' ++ [lst])
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is (tr' ++ [lst])
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
H': ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
  let final_state := finite_trace_last is tr in
  finite_valid_trace XE is tr
  → length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM,
        proper_fixed_equivocator_descriptors
         final_descriptors final_state
        → 
        ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = 
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is (tr' ++ [lst])
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
H': length tr' < length (tr' ++ [lst])
→ ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
    let final_state := finite_trace_last is tr in
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
length tr' < length (tr' ++ [lst]) rewrite app_length.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is (tr' ++ [lst])
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
H': length tr' < length (tr' ++ [lst])
→ ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
    let final_state := finite_trace_last is tr in
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
length tr' < length tr' + length [lst] simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is (tr' ++ [lst])
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
H': length tr' < length (tr' ++ [lst])
→ ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
    let final_state := finite_trace_last is tr in
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
length tr' < length tr' + 1 lia. }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is (tr' ++ [lst])
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
H': ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
  let final_state := finite_trace_last is tr in
  finite_valid_trace XE is tr
  → length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM,
        proper_fixed_equivocator_descriptors
         final_descriptors final_state
        → 
        ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    destruct Htr as [Htr Hinit].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is (tr' ++ [lst])
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
H': ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
  let final_state := finite_trace_last is tr in
  finite_valid_trace XE is tr
  → length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM,
        proper_fixed_equivocator_descriptors
         final_descriptors final_state
        → 
        ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    apply finite_valid_trace_from_app_iff in Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
∧ finite_valid_trace_from XE
    (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
H': ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
  let final_state := finite_trace_last is tr in
  finite_valid_trace XE is tr
  → length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM,
        proper_fixed_equivocator_descriptors
         final_descriptors final_state
        → 
        ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    destruct Htr as [Htr Hlst].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
H': ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
  let final_state := finite_trace_last is tr in
  finite_valid_trace XE is tr
  → length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM,
        proper_fixed_equivocator_descriptors
         final_descriptors final_state
        → 
        ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    specialize (H' tr' (conj Htr Hinit) eq_refl).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    specialize (equivocators_transition_item_project_proper_characterization IM final_descriptors lst) as Hproperx.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hproperx: proper_equivocator_descriptors IM
  final_descriptors 
  (destination lst)
→ ∃ (oitem : 
     option 
       (composite_transition_item IM)) 
    (eqv_descriptors' : 
     equivocator_descriptors IM),
    equivocators_transition_item_project IM
      final_descriptors lst =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
         (∃ Hex : 
         existing_equivocator_label
         (IM (projT1 (l lst)))
         (projT2 (l lst)),
         existT 
         (projT1 (l lst))
         (existing_equivocator_label_extract
         (IM (projT1 (l lst)))
         (projT2 (l lst)) Hex) = 
         l itemx)
         ∧ 
         input lst = input itemx
         ∧ 
         output lst = output itemx
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         destination itemx
         ∧ 
         eqv_descriptors' (projT1 (l lst)) =
         equivocator_label_descriptor
         (projT2 (l lst))
      | None => True
      end
      ∧ (∀ s : 
         composite_state
         (equivocator_IM IM),
         composite_valid
         (equivocator_IM IM) 
         (l lst) 
         (s, input lst)
         → 
         composite_transition
         (equivocator_IM IM) 
         (l lst) 
         (s, input lst) =
         (destination lst, output lst)
         → 
         proper_equivocator_descriptors IM
         eqv_descriptors' s
         ∧ 
         eqv_descriptors' =
         equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l lst))
         (eqv_descriptors' (projT1 (l lst)))
         ∧ 
         s =
         state_update 
         (equivocator_IM IM)
         (destination lst) 
         (projT1 (l lst))
         (s (projT1 (l lst)))
         ∧ 
         previous_state_descriptor_prop
         (IM (projT1 (l lst)))
         (final_descriptors
         (projT1 (l lst)))
         (s (projT1 (l lst)))
         (eqv_descriptors' (projT1 (l lst)))
         ∧ 
         match oitem with
         | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         eqv_descriptors' s
         → 
         composite_valid IM 
         (l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM 
         (l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
         | None =>
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         equivocators_state_project IM
         eqv_descriptors' s
         end)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    specialize
      (equivocators_transition_item_project_preserves_zero_descriptors IM final_descriptors lst)
      as Hzero.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hproperx: proper_equivocator_descriptors IM
  final_descriptors 
  (destination lst)
→ ∃ (oitem : 
     option 
       (composite_transition_item IM)) 
    (eqv_descriptors' : 
     equivocator_descriptors IM),
    equivocators_transition_item_project IM
      final_descriptors lst =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
         (∃ Hex : 
         existing_equivocator_label
         (IM (projT1 (l lst)))
         (projT2 (l lst)),
         existT 
         (projT1 (l lst))
         (existing_equivocator_label_extract
         (IM (projT1 (l lst)))
         (projT2 (l lst)) Hex) = 
         l itemx)
         ∧ 
         input lst = input itemx
         ∧ 
         output lst = output itemx
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         destination itemx
         ∧ 
         eqv_descriptors' (projT1 (l lst)) =
         equivocator_label_descriptor
         (projT2 (l lst))
      | None => True
      end
      ∧ (∀ s : 
         composite_state
         (equivocator_IM IM),
         composite_valid
         (equivocator_IM IM) 
         (l lst) 
         (s, input lst)
         → 
         composite_transition
         (equivocator_IM IM) 
         (l lst) 
         (s, input lst) =
         (destination lst, output lst)
         → 
         proper_equivocator_descriptors IM
         eqv_descriptors' s
         ∧ 
         eqv_descriptors' =
         equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l lst))
         (eqv_descriptors' (projT1 (l lst)))
         ∧ 
         s =
         state_update 
         (equivocator_IM IM)
         (destination lst) 
         (projT1 (l lst))
         (s (projT1 (l lst)))
         ∧ 
         previous_state_descriptor_prop
         (IM (projT1 (l lst)))
         (final_descriptors
         (projT1 (l lst)))
         (s (projT1 (l lst)))
         (eqv_descriptors' (projT1 (l lst)))
         ∧ 
         match oitem with
         | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         eqv_descriptors' s
         → 
         composite_valid IM 
         (l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM 
         (l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
         | None =>
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         equivocators_state_project IM
         eqv_descriptors' s
         end)
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) 
    (s, input lst) =
  (destination lst, output lst)
  → composite_valid 
      (equivocator_IM IM) 
      (l lst) 
      (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    unfold final_state in Hproper.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is tr)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hproperx: proper_equivocator_descriptors IM
  final_descriptors 
  (destination lst)
→ ∃ (oitem : 
     option 
       (composite_transition_item IM)) 
    (eqv_descriptors' : 
     equivocator_descriptors IM),
    equivocators_transition_item_project IM
      final_descriptors lst =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
         (∃ Hex : 
         existing_equivocator_label
         (IM (projT1 (l lst)))
         (projT2 (l lst)),
         existT 
         (projT1 (l lst))
         (existing_equivocator_label_extract
         (IM (projT1 (l lst)))
         (projT2 (l lst)) Hex) = 
         l itemx)
         ∧ 
         input lst = input itemx
         ∧ 
         output lst = output itemx
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         destination itemx
         ∧ 
         eqv_descriptors' (projT1 (l lst)) =
         equivocator_label_descriptor
         (projT2 (l lst))
      | None => True
      end
      ∧ (∀ s : 
         composite_state
         (equivocator_IM IM),
         composite_valid
         (equivocator_IM IM) 
         (l lst) 
         (s, input lst)
         → 
         composite_transition
         (equivocator_IM IM) 
         (l lst) 
         (s, input lst) =
         (destination lst, output lst)
         → 
         proper_equivocator_descriptors IM
         eqv_descriptors' s
         ∧ 
         eqv_descriptors' =
         equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l lst))
         (eqv_descriptors' (projT1 (l lst)))
         ∧ 
         s =
         state_update 
         (equivocator_IM IM)
         (destination lst) 
         (projT1 (l lst))
         (s (projT1 (l lst)))
         ∧ 
         previous_state_descriptor_prop
         (IM (projT1 (l lst)))
         (final_descriptors
         (projT1 (l lst)))
         (s (projT1 (l lst)))
         (eqv_descriptors' (projT1 (l lst)))
         ∧ 
         match oitem with
         | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         eqv_descriptors' s
         → 
         composite_valid IM 
         (l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM 
         (l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
         | None =>
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         equivocators_state_project IM
         eqv_descriptors' s
         end)
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) 
    (s, input lst) =
  (destination lst, output lst)
  → composite_valid 
      (equivocator_IM IM) 
      (l lst) 
      (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    rewrite Htr_lst, finite_trace_last_is_last in Hproper.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (destination lst)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hproperx: proper_equivocator_descriptors IM
  final_descriptors 
  (destination lst)
→ ∃ (oitem : 
     option 
       (composite_transition_item IM)) 
    (eqv_descriptors' : 
     equivocator_descriptors IM),
    equivocators_transition_item_project IM
      final_descriptors lst =
    Some (oitem, eqv_descriptors')
    ∧ match oitem with
      | Some itemx =>
         (∃ Hex : 
         existing_equivocator_label
         (IM (projT1 (l lst)))
         (projT2 (l lst)),
         existT 
         (projT1 (l lst))
         (existing_equivocator_label_extract
         (IM (projT1 (l lst)))
         (projT2 (l lst)) Hex) = 
         l itemx)
         ∧ 
         input lst = input itemx
         ∧ 
         output lst = output itemx
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         destination itemx
         ∧ 
         eqv_descriptors' (projT1 (l lst)) =
         equivocator_label_descriptor
         (projT2 (l lst))
      | None => True
      end
      ∧ (∀ s : 
         composite_state
         (equivocator_IM IM),
         composite_valid
         (equivocator_IM IM) 
         (l lst) 
         (s, input lst)
         → 
         composite_transition
         (equivocator_IM IM) 
         (l lst) 
         (s, input lst) =
         (destination lst, output lst)
         → 
         proper_equivocator_descriptors IM
         eqv_descriptors' s
         ∧ 
         eqv_descriptors' =
         equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l lst))
         (eqv_descriptors' (projT1 (l lst)))
         ∧ 
         s =
         state_update 
         (equivocator_IM IM)
         (destination lst) 
         (projT1 (l lst))
         (s (projT1 (l lst)))
         ∧ 
         previous_state_descriptor_prop
         (IM (projT1 (l lst)))
         (final_descriptors
         (projT1 (l lst)))
         (s (projT1 (l lst)))
         (eqv_descriptors' (projT1 (l lst)))
         ∧ 
         match oitem with
         | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         eqv_descriptors' s
         → 
         composite_valid IM 
         (l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM 
         (l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
         | None =>
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         equivocators_state_project IM
         eqv_descriptors' s
         end)
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) 
    (s, input lst) =
  (destination lst, output lst)
  → composite_valid 
      (equivocator_IM IM) 
      (l lst) 
      (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    spec Hproperx (proj1 Hproper).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (destination lst)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) 
    (s, input lst) =
  (destination lst, output lst)
  → composite_valid 
      (equivocator_IM IM) 
      (l lst) 
      (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
Hproperx: ∃ (oitem : 
   option 
     (composite_transition_item IM)) 
  (eqv_descriptors' : 
   equivocator_descriptors IM),
  equivocators_transition_item_project IM
    final_descriptors lst =
  Some (oitem, eqv_descriptors')
  ∧ match oitem with
    | Some itemx =>
        (∃ Hex : 
         existing_equivocator_label
         (IM (projT1 (l lst)))
         (projT2 (l lst)),
         existT 
         (projT1 (l lst))
         (existing_equivocator_label_extract
         (IM (projT1 (l lst)))
         (projT2 (l lst)) Hex) = 
         l itemx)
        ∧ 
        input lst = input itemx
        ∧ 
        output lst = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (destination lst) =
        destination itemx
        ∧ 
        eqv_descriptors' (projT1 (l lst)) =
        equivocator_label_descriptor
         (projT2 (l lst))
    | None => True
    end
    ∧ (∀ s : 
         composite_state
         (equivocator_IM IM),
         composite_valid
         (equivocator_IM IM) 
         (l lst) 
         (s, input lst)
         → 
         composite_transition
         (equivocator_IM IM) 
         (l lst) 
         (s, input lst) =
         (destination lst, output lst)
         → 
         proper_equivocator_descriptors IM
         eqv_descriptors' s
         ∧ 
         eqv_descriptors' =
         equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l lst))
         (eqv_descriptors' (projT1 (l lst)))
         ∧ 
         s =
         state_update 
         (equivocator_IM IM)
         (destination lst) 
         (projT1 (l lst))
         (s (projT1 (l lst)))
         ∧ 
         previous_state_descriptor_prop
         (IM (projT1 (l lst)))
         (final_descriptors
         (projT1 (l lst)))
         (s (projT1 (l lst)))
         (eqv_descriptors' (projT1 (l lst)))
         ∧ 
         match oitem with
         | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         eqv_descriptors' s
         → 
         composite_valid IM 
         (l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM 
         (l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
         | None =>
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         equivocators_state_project IM
         eqv_descriptors' s
         end)
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    destruct Hproperx as [oitem [final_descriptors' [Hprojectx [Hitemx Hproperx]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (destination lst)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) 
    (s, input lst) =
  (destination lst, output lst)
  → composite_valid 
      (equivocator_IM IM) 
      (l lst) 
      (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 (l lst)))
         (projT2 (l lst)),
       existT 
         (projT1 (l lst))
         (existing_equivocator_label_extract
         (IM (projT1 (l lst)))
         (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors 
         (destination lst) =
        destination itemx
        ∧ 
        final_descriptors' (projT1 (l lst)) =
        equivocator_label_descriptor
         (projT2 (l lst))
| None => True
end
Hproperx: ∀ s : composite_state (equivocator_IM IM),
  composite_valid 
    (equivocator_IM IM) 
    (l lst) 
    (s, input lst)
  → composite_transition
      (equivocator_IM IM) 
      (l lst) 
      (s, input lst) =
    (destination lst, output lst)
    → proper_equivocator_descriptors IM
        final_descriptors' s
      ∧ final_descriptors' =
        equivocator_descriptors_update IM
         final_descriptors 
         (projT1 (l lst))
         (final_descriptors'
         (projT1 (l lst)))
        ∧ 
        s =
        state_update 
         (equivocator_IM IM)
         (destination lst) 
         (projT1 (l lst))
         (s (projT1 (l lst)))
        ∧ 
        previous_state_descriptor_prop
         (IM (projT1 (l lst)))
         (final_descriptors
         (projT1 (l lst)))
         (s (projT1 (l lst)))
         (final_descriptors'
         (projT1 (l lst)))
        ∧ 
        match oitem with
        | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         final_descriptors' s
         → 
         composite_valid IM 
         (l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM 
         (l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
        | None =>
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         equivocators_state_project IM
         final_descriptors' s
        end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    specialize (Hproperx (finite_trace_last is tr')).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (destination lst)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) 
    (s, input lst) =
  (destination lst, output lst)
  → composite_valid 
      (equivocator_IM IM) 
      (l lst) 
      (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 (l lst)))
         (projT2 (l lst)),
       existT 
         (projT1 (l lst))
         (existing_equivocator_label_extract
         (IM (projT1 (l lst)))
         (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors 
         (destination lst) =
        destination itemx
        ∧ 
        final_descriptors' (projT1 (l lst)) =
        equivocator_label_descriptor
         (projT2 (l lst))
| None => True
end
Hproperx: composite_valid 
  (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
         (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update 
         (equivocator_IM IM)
         (destination lst) 
         (projT1 (l lst))
         (finite_trace_last is tr'
         (projT1 (l lst)))
        ∧ 
        previous_state_descriptor_prop
         (IM (projT1 (l lst)))
         (final_descriptors
         (projT1 (l lst)))
         (finite_trace_last is tr'
         (projT1 (l lst)))
         (final_descriptors'
         (projT1 (l lst)))
        ∧ 
        match oitem with
        | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
         → 
         composite_valid IM 
         (l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM 
         (l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
        | None =>
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
        end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors
      (tr' ++ [lst]) = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X' isX trX
    unfold equivocators_trace_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (destination lst)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) 
    (s, input lst) =
  (destination lst, output lst)
  → composite_valid 
      (equivocator_IM IM) 
      (l lst) 
      (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 (l lst)))
         (projT2 (l lst)),
       existT 
         (projT1 (l lst))
         (existing_equivocator_label_extract
         (IM (projT1 (l lst)))
         (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors 
         (destination lst) =
        destination itemx
        ∧ 
        final_descriptors' (projT1 (l lst)) =
        equivocator_label_descriptor
         (projT2 (l lst))
| None => True
end
Hproperx: composite_valid 
  (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
         (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update 
         (equivocator_IM IM)
         (destination lst) 
         (projT1 (l lst))
         (finite_trace_last is tr'
         (projT1 (l lst)))
        ∧ 
        previous_state_descriptor_prop
         (IM (projT1 (l lst)))
         (final_descriptors
         (projT1 (l lst)))
         (finite_trace_last is tr'
         (projT1 (l lst)))
         (final_descriptors'
         (projT1 (l lst)))
        ∧ 
        match oitem with
        | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
         → 
         composite_valid IM 
         (l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM 
         (l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
        | None =>
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
        end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM)
      (Some ([], final_descriptors)) (tr' ++ [lst]) =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    rewrite fold_right_app.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (destination lst)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) 
    (s, input lst) =
  (destination lst, output lst)
  → composite_valid 
      (equivocator_IM IM) 
      (l lst) 
      (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 (l lst)))
         (projT2 (l lst)),
       existT 
         (projT1 (l lst))
         (existing_equivocator_label_extract
         (IM (projT1 (l lst)))
         (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors 
         (destination lst) =
        destination itemx
        ∧ 
        final_descriptors' (projT1 (l lst)) =
        equivocator_label_descriptor
         (projT2 (l lst))
| None => True
end
Hproperx: composite_valid 
  (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
         (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update 
         (equivocator_IM IM)
         (destination lst) 
         (projT1 (l lst))
         (finite_trace_last is tr'
         (projT1 (l lst)))
        ∧ 
        previous_state_descriptor_prop
         (IM (projT1 (l lst)))
         (final_descriptors
         (projT1 (l lst)))
         (finite_trace_last is tr'
         (projT1 (l lst)))
         (final_descriptors'
         (projT1 (l lst)))
        ∧ 
        match oitem with
        | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
         → 
         composite_valid IM 
         (l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM 
         (l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
        | None =>
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
        end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM)
      (foldr (equivocators_trace_project_folder IM)
         (Some ([], final_descriptors)) [lst]) tr' =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    match goal with
    |- context [fold_right _ ?fld _] => remember fld as foldx
    end.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (destination lst)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) 
    (s, input lst) =
  (destination lst, output lst)
  → composite_valid 
      (equivocator_IM IM) 
      (l lst) 
      (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 (l lst)))
         (projT2 (l lst)),
       existT 
         (projT1 (l lst))
         (existing_equivocator_label_extract
         (IM (projT1 (l lst)))
         (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors 
         (destination lst) =
        destination itemx
        ∧ 
        final_descriptors' (projT1 (l lst)) =
        equivocator_label_descriptor
         (projT2 (l lst))
| None => True
end
Hproperx: composite_valid 
  (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
         (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update 
         (equivocator_IM IM)
         (destination lst) 
         (projT1 (l lst))
         (finite_trace_last is tr'
         (projT1 (l lst)))
        ∧ 
        previous_state_descriptor_prop
         (IM (projT1 (l lst)))
         (final_descriptors
         (projT1 (l lst)))
         (finite_trace_last is tr'
         (projT1 (l lst)))
         (final_descriptors'
         (projT1 (l lst)))
        ∧ 
        match oitem with
        | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
         → 
         composite_valid IM 
         (l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM 
         (l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
        | None =>
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
        end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
foldr
  (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors)) [lst]
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    simpl in Heqfoldx.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (destination lst)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) 
    (s, input lst) =
  (destination lst, output lst)
  → composite_valid 
      (equivocator_IM IM) 
      (l lst) 
      (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 (l lst)))
         (projT2 (l lst)),
       existT 
         (projT1 (l lst))
         (existing_equivocator_label_extract
         (IM (projT1 (l lst)))
         (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors 
         (destination lst) =
        destination itemx
        ∧ 
        final_descriptors' (projT1 (l lst)) =
        equivocator_label_descriptor
         (projT2 (l lst))
| None => True
end
Hproperx: composite_valid 
  (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
         (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update 
         (equivocator_IM IM)
         (destination lst) 
         (projT1 (l lst))
         (finite_trace_last is tr'
         (projT1 (l lst)))
        ∧ 
        previous_state_descriptor_prop
         (IM (projT1 (l lst)))
         (final_descriptors
         (projT1 (l lst)))
         (finite_trace_last is tr'
         (projT1 (l lst)))
         (final_descriptors'
         (projT1 (l lst)))
        ∧ 
        match oitem with
        | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
         → 
         composite_valid IM 
         (l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM 
         (l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
        | None =>
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
        end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match
  equivocators_transition_item_project IM
    final_descriptors lst
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    rewrite Hprojectx in Heqfoldx.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: finite_valid_trace_from XE
  (finite_trace_last is tr') [lst]
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (destination lst)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) 
    (s, input lst) =
  (destination lst, output lst)
  → composite_valid 
      (equivocator_IM IM) 
      (l lst) 
      (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 (l lst)))
         (projT2 (l lst)),
       existT 
         (projT1 (l lst))
         (existing_equivocator_label_extract
         (IM (projT1 (l lst)))
         (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors 
         (destination lst) =
        destination itemx
        ∧ 
        final_descriptors' (projT1 (l lst)) =
        equivocator_label_descriptor
         (projT2 (l lst))
| None => True
end
Hproperx: composite_valid 
  (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
         (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update 
         (equivocator_IM IM)
         (destination lst) 
         (projT1 (l lst))
         (finite_trace_last is tr'
         (projT1 (l lst)))
        ∧ 
        previous_state_descriptor_prop
         (IM (projT1 (l lst)))
         (final_descriptors
         (projT1 (l lst)))
         (finite_trace_last is tr'
         (projT1 (l lst)))
         (final_descriptors'
         (projT1 (l lst)))
        ∧ 
        match oitem with
        | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
         → 
         composite_valid IM 
         (l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM 
         (l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
        | None =>
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
        end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    apply first_transition_valid in Hlst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
lst: composite_transition_item (equivocator_IM IM)
H: ∀ y : nat,
  y < length (tr' ++ [lst])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: input_valid_transition XE 
  (l lst) 
  (finite_trace_last is tr', input lst)
  (destination lst, output lst)
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (destination lst)
Htr_lst: tr = tr' ++ [lst]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM) 
    (l lst) 
    (s, input lst) =
  (destination lst, output lst)
  → composite_valid 
      (equivocator_IM IM) 
      (l lst) 
      (s, input lst)
    → equivocators_transition_item_project IM
        final_descriptors lst =
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors lst =
Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 (l lst)))
         (projT2 (l lst)),
       existT 
         (projT1 (l lst))
         (existing_equivocator_label_extract
         (IM (projT1 (l lst)))
         (projT2 (l lst)) Hex) = 
       l itemx)
    ∧ input lst = input itemx
      ∧ output lst = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors 
         (destination lst) =
        destination itemx
        ∧ 
        final_descriptors' (projT1 (l lst)) =
        equivocator_label_descriptor
         (projT2 (l lst))
| None => True
end
Hproperx: composite_valid 
  (equivocator_IM IM) 
  (l lst)
  (finite_trace_last is tr', input lst)
→ composite_transition 
    (equivocator_IM IM) 
    (l lst)
    (finite_trace_last is tr', input lst) =
  (destination lst, output lst)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 (l lst))
        (final_descriptors'
         (projT1 (l lst)))
      ∧ finite_trace_last is tr' =
        state_update 
         (equivocator_IM IM)
         (destination lst) 
         (projT1 (l lst))
         (finite_trace_last is tr'
         (projT1 (l lst)))
        ∧ 
        previous_state_descriptor_prop
         (IM (projT1 (l lst)))
         (final_descriptors
         (projT1 (l lst)))
         (finite_trace_last is tr'
         (projT1 (l lst)))
         (final_descriptors'
         (projT1 (l lst)))
        ∧ 
        match oitem with
        | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
         → 
         composite_valid IM 
         (l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM 
         (l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
        | None =>
         equivocators_state_project IM
         final_descriptors
         (destination lst) =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
        end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX destruct lst as (l, iom, s, oom).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: label
iom: option message
s: state
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      let final_state := 
        finite_trace_last is tr in
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors final_state
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         let isX :=
         equivocators_state_project IM
         initial_descriptors is in
         let final_stateX :=
         finite_trace_last isX trX in
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors final_state =
         final_stateX
         ∧ 
         finite_valid_trace X' isX trX
final_state:= finite_trace_last is tr: state
Htr: finite_valid_trace_from XE is tr'
Hlst: input_valid_transition XE
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (finite_trace_last is tr',
  input
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |})
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
  output
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |})
Hinit: initial_state_prop is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      let isX :=
        equivocators_state_project IM
         initial_descriptors is in
      let final_stateX :=
        finite_trace_last isX trX in
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        final_stateX
        ∧ 
        finite_valid_trace X' isX trX
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s0 : composite_state (equivocator_IM IM)),
  composite_transition 
    (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (s0,
    input
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}) =
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
  output
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |})
  → composite_valid 
      (equivocator_IM IM)
      (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})
      (s0,
      input
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |})
    → equivocators_transition_item_project IM
        final_descriptors
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |} = 
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
         (projT2
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})),
       existT
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}))
         (existing_equivocator_label_extract
         (IM
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
         (projT2
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})) Hex) = 
       VLSM.l itemx)
    ∧ input
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |} = 
      input itemx
      ∧ output
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |} = 
        output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (destination
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}) = 
        destination itemx
        ∧ 
        final_descriptors'
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})) =
        equivocator_label_descriptor
         (projT2
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}))
| None => True
end
Hproperx: composite_valid 
  (equivocator_IM IM)
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (finite_trace_last is tr',
  input
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |})
→ composite_transition 
    (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (finite_trace_last is tr',
    input
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}) =
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
  output
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |})
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors
        (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}))
        (final_descriptors'
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
      ∧ finite_trace_last is tr' =
        state_update 
         (equivocator_IM IM)
         (destination
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}))
         (finite_trace_last is tr'
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
        ∧ 
        previous_state_descriptor_prop
         (IM
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
         (final_descriptors
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
         (finite_trace_last is tr'
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
         (final_descriptors'
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
        ∧ 
        match oitem with
        | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
         → 
         composite_valid IM 
         (VLSM.l itemx) 
         (sx, input itemx)
         ∧ 
         composite_transition IM
         (VLSM.l itemx) 
         (sx, input itemx) =
         (destination itemx, output itemx)
        | None =>
         equivocators_state_project IM
         final_descriptors
         (destination
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}) =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
        end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX simpl in *.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hlst: input_valid_transition XE l
  (finite_trace_last is tr', iom) 
  (s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s0 : composite_state (equivocator_IM IM)),
  (let (i, li) := l in
   let (si', om') :=
     vtransition 
       (equivocator_IM IM i) li 
       (s0 i, iom) in
   (state_update (equivocator_IM IM) s0 i si',
   om')) = 
  (s, oom)
  → (let (i, li) := l in
     vvalid 
       (equivocator_IM IM i) li 
       (s0 i, iom))
    → equivocators_transition_item_project IM
        final_descriptors
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |} = 
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hproperx: (let (i, li) := l in
 vvalid (equivocator_IM IM i) li
   (finite_trace_last is tr' i, iom))
→ (let (i, li) := l in
   let (si', om') :=
     vtransition 
       (equivocator_IM IM i) li
       (finite_trace_last is tr' i, iom) in
   (state_update 
      (equivocator_IM IM)
      (finite_trace_last is tr') i si',
   om')) = 
  (s, oom)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l)
        (final_descriptors' (projT1 l))
      ∧ finite_trace_last is tr' =
        state_update 
         (equivocator_IM IM) s 
         (projT1 l)
         (finite_trace_last is tr'
         (projT1 l))
        ∧ 
        previous_state_descriptor_prop
         (IM (projT1 l))
         (final_descriptors (projT1 l))
         (finite_trace_last is tr'
         (projT1 l))
         (final_descriptors' (projT1 l))
        ∧ 
        match oitem with
        | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
         → 
         (let (i, li) := VLSM.l itemx in
         vvalid 
         (IM i) li 
         (sx i, input itemx))
         ∧ 
         (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
         (destination itemx, output itemx)
        | None =>
         equivocators_state_project IM
         final_descriptors s =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
        end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    destruct Hlst as [[Hs [Hiom [Hv Hc]]] Ht].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
Hzero: ∀ (oitem : 
   option 
     (composite_transition_item IM)) 
  (idescriptors : 
   equivocator_descriptors IM) 
  (s0 : composite_state (equivocator_IM IM)),
  (let (i, li) := l in
   let (si', om') :=
     vtransition 
       (equivocator_IM IM i) li 
       (s0 i, iom) in
   (state_update (equivocator_IM IM) s0 i si',
   om')) = 
  (s, oom)
  → (let (i, li) := l in
     vvalid 
       (equivocator_IM IM i) li 
       (s0 i, iom))
    → equivocators_transition_item_project IM
        final_descriptors
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |} = 
      Some (oitem, idescriptors)
      → ∀ i : index,
         final_descriptors i = Existing 0
         → 
         idescriptors i = Existing 0
oitem: option (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hproperx: (let (i, li) := l in
 vvalid (equivocator_IM IM i) li
   (finite_trace_last is tr' i, iom))
→ (let (i, li) := l in
   let (si', om') :=
     vtransition 
       (equivocator_IM IM i) li
       (finite_trace_last is tr' i, iom) in
   (state_update 
      (equivocator_IM IM)
      (finite_trace_last is tr') i si',
   om')) = 
  (s, oom)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l)
        (final_descriptors' (projT1 l))
      ∧ finite_trace_last is tr' =
        state_update 
         (equivocator_IM IM) s 
         (projT1 l)
         (finite_trace_last is tr'
         (projT1 l))
        ∧ 
        previous_state_descriptor_prop
         (IM (projT1 l))
         (final_descriptors (projT1 l))
         (finite_trace_last is tr'
         (projT1 l))
         (final_descriptors' (projT1 l))
        ∧ 
        match oitem with
        | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
         → 
         (let (i, li) := VLSM.l itemx in
         vvalid 
         (IM i) li 
         (sx i, input itemx))
         ∧ 
         (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
         (destination itemx, output itemx)
        | None =>
         equivocators_state_project IM
         final_descriptors s =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
        end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    specialize (Hzero oitem final_descriptors' _ Ht Hv Hprojectx).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hproperx: (let (i, li) := l in
 vvalid (equivocator_IM IM i) li
   (finite_trace_last is tr' i, iom))
→ (let (i, li) := l in
   let (si', om') :=
     vtransition 
       (equivocator_IM IM i) li
       (finite_trace_last is tr' i, iom) in
   (state_update 
      (equivocator_IM IM)
      (finite_trace_last is tr') i si',
   om')) = 
  (s, oom)
  → proper_equivocator_descriptors IM
      final_descriptors'
      (finite_trace_last is tr')
    ∧ final_descriptors' =
      equivocator_descriptors_update IM
        final_descriptors 
        (projT1 l)
        (final_descriptors' (projT1 l))
      ∧ finite_trace_last is tr' =
        state_update 
         (equivocator_IM IM) s 
         (projT1 l)
         (finite_trace_last is tr'
         (projT1 l))
        ∧ 
        previous_state_descriptor_prop
         (IM (projT1 l))
         (final_descriptors (projT1 l))
         (finite_trace_last is tr'
         (projT1 l))
         (final_descriptors' (projT1 l))
        ∧ 
        match oitem with
        | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
         → 
         (let (i, li) := VLSM.l itemx in
         vvalid 
         (IM i) li 
         (sx i, input itemx))
         ∧ 
         (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
         (destination itemx, output itemx)
        | None =>
         equivocators_state_project IM
         final_descriptors s =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
        end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    specialize (Hproperx Hv Ht).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hproperx: proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
∧ final_descriptors' =
  equivocator_descriptors_update IM
    final_descriptors 
    (projT1 l)
    (final_descriptors' (projT1 l))
  ∧ finite_trace_last is tr' =
    state_update 
      (equivocator_IM IM) s 
      (projT1 l)
      (finite_trace_last is tr' (projT1 l))
    ∧ previous_state_descriptor_prop
        (IM (projT1 l))
        (final_descriptors (projT1 l))
        (finite_trace_last is tr'
         (projT1 l))
        (final_descriptors' (projT1 l))
      ∧ match oitem with
        | Some itemx =>
         ∀ sx : composite_state IM,
         sx =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
         → 
         (let (i, li) := VLSM.l itemx in
         vvalid 
         (IM i) li 
         (sx i, input itemx))
         ∧ 
         (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
         (destination itemx, output itemx)
        | None =>
         equivocators_state_project IM
         final_descriptors s =
         equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')
        end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    destruct Hproperx as [_Hproper' [_ [_ [_ Hx]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    assert (Hproper' : proper_fixed_equivocator_descriptors final_descriptors' (finite_trace_last is tr')).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
proper_fixed_equivocator_descriptors
  final_descriptors' (finite_trace_last is tr')
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr 
      (equivocators_trace_project_folder IM) foldx tr' =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX
      ∧ finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
    {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
proper_fixed_equivocator_descriptors
  final_descriptors' (finite_trace_last is tr') split; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
∀ i : index,
  i ∉ equivocating → final_descriptors' i = Existing 0
      intros i Hi.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
i: index
Hi: i ∉ equivocating
final_descriptors' i = Existing 0 apply Hzero.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
i: index
Hi: i ∉ equivocating
final_descriptors i = Existing 0 clear Hzero.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
i: index
Hi: i ∉ equivocating
final_descriptors i = Existing 0 destruct Hproper as [_ Hzero].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hzero: ∀ i : index,
  i ∉ equivocating
  → final_descriptors i = Existing 0
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
i: index
Hi: i ∉ equivocating
final_descriptors i = Existing 0
      apply Hzero.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hzero: ∀ i : index,
  i ∉ equivocating
  → final_descriptors i = Existing 0
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
i: index
Hi: i ∉ equivocating
i ∉ equivocating assumption.
    }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
Hzero: ∀ i : index,
  final_descriptors i = Existing 0
  → final_descriptors' i = Existing 0
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
_Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is tr')
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    clear _Hproper' Hzero.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
H': ∀ final_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    final_descriptors 
    (finite_trace_last is tr')
  → ∃ (trX : list (composite_transition_item IM)) 
      (initial_descriptors : 
       equivocator_descriptors IM),
      proper_fixed_equivocator_descriptors
        initial_descriptors is
      ∧ equivocators_trace_project IM
         final_descriptors tr' =
        Some (trX, initial_descriptors)
        ∧ 
        equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr') =
        finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
        ∧ 
        finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_descriptors': equivocator_descriptors IM
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    specialize (H' _ Hproper').message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
H': ∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM
      final_descriptors' tr' =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') =
      finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX
      ∧ finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    destruct H' as [trX' [initial_descriptors [_ [Htr_project [Hstate_project HtrX']]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    assert
      (Htr' : finite_valid_trace FreeE is tr').message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
finite_valid_trace FreeE is tr'
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr 
      (equivocators_trace_project_folder IM) foldx tr' =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX
      ∧ finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
    {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
finite_valid_trace FreeE is tr'  split; [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
finite_valid_trace_from FreeE is tr'
      revert Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
finite_valid_trace_from XE is tr'
→ finite_valid_trace_from FreeE is tr'  apply VLSM_incl_finite_valid_trace_from.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
VLSM_incl_part
  (composite_vlsm_machine (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint IM
        Hbs index_listing equivocating))
  (composite_vlsm_machine (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)))
      apply equivocators_fixed_equivocations_vlsm_incl_free.
    }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    assert
      (Htr'pre : finite_valid_trace (pre_loaded_with_all_messages_vlsm FreeE) is tr').message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE) is tr'
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr 
      (equivocators_trace_project_folder IM) foldx tr' =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX
      ∧ finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
    {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE) is tr' split; [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) is tr'
      specialize (vlsm_incl_pre_loaded_with_all_messages_vlsm FreeE) as Hincl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Hincl: VLSM_incl FreeE
  (pre_loaded_with_all_messages_vlsm FreeE)
finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) is tr'
      apply (VLSM_incl_finite_valid_trace_from Hincl).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Hincl: VLSM_incl FreeE
  (pre_loaded_with_all_messages_vlsm FreeE)
finite_valid_trace_from (mk_vlsm (machine FreeE)) is
  tr' apply Htr'.
    }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    specialize
      (equivocators_trace_project_preserves_proper_fixed_equivocator_descriptors _ _ (proj1 Htr'pre) _ _ _ Htr_project Hproper')
      as Hproper_initial.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
oitem: option (composite_transition_item IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (oitem, final_descriptors')
Hitemx: match oitem with
| Some itemx =>
    (∃ Hex : 
        existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
       existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex) = 
       VLSM.l itemx)
    ∧ iom = input itemx
      ∧ oom = output itemx
        ∧ 
        equivocators_state_project IM
         final_descriptors s =
        destination itemx
        ∧ 
        final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
| None => True
end
Hx: match oitem with
| Some itemx =>
    ∀ sx : composite_state IM,
      sx =
      equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr')
      → (let (i, li) := VLSM.l itemx in
         vvalid (IM i) li (sx i, input itemx))
        ∧ 
        (let (i, li) := VLSM.l itemx in
         let 
         (si', om') :=
         vtransition 
         (IM i) li 
         (sx i, input itemx) in
         (state_update IM sx i si', om')) =
        (destination itemx, output itemx)
| None =>
    equivocators_state_project IM
      final_descriptors s =
    equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr')
end
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx =
match oitem with
| Some item' =>
    Some ([item'], final_descriptors')
| None => Some ([], final_descriptors')
end
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
    destruct oitem as [item|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hitemx: (∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
   existT 
     (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
Hx: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors' 
    (finite_trace_last is tr')
  → (let (i, li) := VLSM.l item in
     vvalid (IM i) li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let 
         (si', om') :=
         vtransition (IM i) li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr 
      (equivocators_trace_project_folder IM) foldx tr' =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX
      ∧ finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
    +message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hitemx: (∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
   existT 
     (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
Hx: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors' 
    (finite_trace_last is tr')
  → (let (i, li) := VLSM.l item in
     vvalid (IM i) li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let 
         (si', om') :=
         vtransition (IM i) li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX  simpl in Hitemx.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hitemx: (∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
   existT 
     (projT1 l)
     (existing_equivocator_label_extract
        (IM (projT1 l)) 
        (projT2 l) Hex) = 
   VLSM.l item)
∧ iom = input item
  ∧ oom = output item
    ∧ equivocators_state_project IM
        final_descriptors s =
      destination item
      ∧ final_descriptors' (projT1 l) =
        equivocator_label_descriptor
         (projT2 l)
Hx: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors' 
    (finite_trace_last is tr')
  → (let (i, li) := VLSM.l item in
     vvalid (IM i) li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let 
         (si', om') :=
         vtransition (IM i) li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX destruct Hitemx as [Hl [Hinput [Houtput [Hdestination _]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hx: ∀ sx : composite_state IM,
  sx =
  equivocators_state_project IM
    final_descriptors' 
    (finite_trace_last is tr')
  → (let (i, li) := VLSM.l item in
     vvalid (IM i) li (sx i, input item))
    ∧ (let (i, li) := VLSM.l item in
       let 
         (si', om') :=
         vtransition (IM i) li (sx i, input item) in
       (state_update IM sx i si', om')) =
      (destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
      specialize (Hx _ eq_refl).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hx: (let (i, li) := VLSM.l item in
 vvalid (IM i) li
   (equivocators_state_project IM
      final_descriptors'
      (finite_trace_last is tr') i, 
   input item))
∧ (let (i, li) := VLSM.l item in
   let (si', om') :=
     vtransition 
       (IM i) li
       (equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr') i,
       input item) in
   (state_update IM
      (equivocators_state_project IM
         final_descriptors'
         (finite_trace_last is tr')) i si', om')) =
  (destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
      destruct Hx as [Hvx Htx].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX
      exists (trX' ++ [item]), initial_descriptors.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([item], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
proper_fixed_equivocator_descriptors
  initial_descriptors is
∧ foldr (equivocators_trace_project_folder IM) foldx
    tr' = Some (trX' ++ [item], initial_descriptors)
  ∧ equivocators_state_project IM final_descriptors
      final_state =
    finite_trace_last
      (equivocators_state_project IM
         initial_descriptors is) (trX' ++ [item])
    ∧ finite_valid_trace X'
        (equivocators_state_project IM
           initial_descriptors is) (trX' ++ [item]) subst foldx.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
proper_fixed_equivocator_descriptors
  initial_descriptors is
∧ foldr (equivocators_trace_project_folder IM)
    (Some ([item], final_descriptors')) tr' =
  Some (trX' ++ [item], initial_descriptors)
  ∧ equivocators_state_project IM final_descriptors
      final_state =
    finite_trace_last
      (equivocators_state_project IM
         initial_descriptors is) (trX' ++ [item])
    ∧ finite_valid_trace X'
        (equivocators_state_project IM
           initial_descriptors is) (trX' ++ [item])
      rewrite equivocators_trace_project_folder_additive with (trX := trX') (eqv_descriptors := initial_descriptors)
      ; [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
proper_fixed_equivocator_descriptors
  initial_descriptors is
∧ Some (trX' ++ [item], initial_descriptors) =
  Some (trX' ++ [item], initial_descriptors)
  ∧ equivocators_state_project IM final_descriptors
      final_state =
    finite_trace_last
      (equivocators_state_project IM
         initial_descriptors is) (trX' ++ [item])
    ∧ finite_valid_trace X'
        (equivocators_state_project IM
           initial_descriptors is) (trX' ++ [item])
      split; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Some (trX' ++ [item], initial_descriptors) =
Some (trX' ++ [item], initial_descriptors)
∧ equivocators_state_project IM final_descriptors
    final_state =
  finite_trace_last
    (equivocators_state_project IM initial_descriptors
       is) (trX' ++ [item])
  ∧ finite_valid_trace X'
      (equivocators_state_project IM
         initial_descriptors is) (trX' ++ [item])
      split; [reflexivity|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM initial_descriptors
     is) (trX' ++ [item])
∧ finite_valid_trace X'
    (equivocators_state_project IM initial_descriptors
       is) (trX' ++ [item])
      rewrite finite_trace_last_is_last.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  final_state = destination item
∧ finite_valid_trace X'
    (equivocators_state_project IM initial_descriptors
       is) (trX' ++ [item])
      unfold final_state.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  (finite_trace_last is tr) = destination item
∧ finite_valid_trace X'
    (equivocators_state_project IM initial_descriptors
       is) (trX' ++ [item]) subst tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  (finite_trace_last is
     (tr' ++
      [{|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |}])) = destination item
∧ finite_valid_trace X'
    (equivocators_state_project IM initial_descriptors
       is) (trX' ++ [item])
      rewrite finite_trace_last_is_last.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) = destination item
∧ finite_valid_trace X'
    (equivocators_state_project IM initial_descriptors
       is) (trX' ++ [item])
      split; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
finite_valid_trace X'
  (equivocators_state_project IM initial_descriptors
     is) (trX' ++ [item])
      destruct HtrX' as [HtrX' His].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
finite_valid_trace X'
  (equivocators_state_project IM initial_descriptors
     is) (trX' ++ [item])
      split; [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
finite_valid_trace_from X'
  (equivocators_state_project IM initial_descriptors
     is) (trX' ++ [item])
      apply finite_valid_trace_from_app_iff.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
finite_valid_trace_from X'
  (equivocators_state_project IM initial_descriptors
     is) trX'
∧ finite_valid_trace_from X'
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX') [item]
      split; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
finite_valid_trace_from X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX') [item]
      change [item] with ([] ++ [item]).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
finite_valid_trace_from X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX') ([] ++ [item])
      match goal with
      |- finite_valid_trace_from _ ?l _ => remember l as lst
      end.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
item: composite_transition_item IM
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some (Some item, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = 
  VLSM.l item
Hinput: iom = input item
Houtput: oom = output item
Hdestination: equivocators_state_project IM
  final_descriptors s =
destination item
Hvx: let (i, li) := VLSM.l item in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, 
  input item)
Htx: (let (i, li) := VLSM.l item in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     input item) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination item, output item)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
finite_valid_trace_from X' lst ([] ++ [item])
      destruct item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
l0: label
input: option message
destination: state
output: option message
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) =
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}
Hinput: iom =
VLSM.input
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Houtput: oom =
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     VLSM.input
       {|
         l := l0;
         input := input;
         destination := destination;
         output := output
       |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
finite_valid_trace_from X' lst
  ([] ++
   [{|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}])
      assert (Hplst : valid_state_prop X' lst).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
l0: label
input: option message
destination: state
output: option message
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) =
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}
Hinput: iom =
VLSM.input
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Houtput: oom =
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     VLSM.input
       {|
         l := l0;
         input := input;
         destination := destination;
         output := output
       |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
valid_state_prop X' lst
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
l0: label
input: option message
destination: state
output: option message
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) =
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}
Hinput: iom =
VLSM.input
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Houtput: oom =
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     VLSM.input
       {|
         l := l0;
         input := input;
         destination := destination;
         output := output
       |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hplst: valid_state_prop X' lst
finite_valid_trace_from X' lst
  ([] ++
   [{|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}])
      {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
l0: label
input: option message
destination: state
output: option message
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) =
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}
Hinput: iom =
VLSM.input
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Houtput: oom =
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     VLSM.input
       {|
         l := l0;
         input := input;
         destination := destination;
         output := output
       |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
valid_state_prop X' lst apply finite_valid_trace_last_pstate in HtrX'.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
l0: label
input: option message
destination: state
output: option message
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) =
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}
Hinput: iom =
VLSM.input
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Houtput: oom =
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     VLSM.input
       {|
         l := l0;
         input := input;
         destination := destination;
         output := output
       |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
valid_state_prop X' lst subst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: option message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) =
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     VLSM.input
       {|
         l := l0;
         input := input;
         destination := destination;
         output := output
       |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX') assumption. }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
l0: label
input: option message
destination: state
output: option message
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) =
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}
Hinput: iom =
VLSM.input
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Houtput: oom =
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     VLSM.input
       {|
         l := l0;
         input := input;
         destination := destination;
         output := output
       |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hplst: valid_state_prop X' lst
finite_valid_trace_from X' lst
  ([] ++
   [{|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}])
      apply (extend_right_finite_trace_from X'); [constructor; assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
l0: label
input: option message
destination: state
output: option message
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) =
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}
Hinput: iom =
VLSM.input
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Houtput: oom =
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     VLSM.input
       {|
         l := l0;
         input := input;
         destination := destination;
         output := output
       |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hplst: valid_state_prop X' lst
input_valid_transition X' l0
  (finite_trace_last lst [], input)
  (destination, output)
      simpl in Hl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
l0: label
input: option message
destination: state
output: option message
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hinput: iom =
VLSM.input
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Houtput: oom =
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     VLSM.input
       {|
         l := l0;
         input := input;
         destination := destination;
         output := output
       |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
lst: state
Heqlst: lst =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hplst: valid_state_prop X' lst
input_valid_transition X' l0
  (finite_trace_last lst [], input)
  (destination, output) subst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: option message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     VLSM.input
       {|
         l := l0;
         input := input;
         destination := destination;
         output := output
       |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [], input)
  (destination, output)
      simpl in Hc.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: option message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', input)
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     VLSM.input
       {|
         l := l0;
         input := input;
         destination := destination;
         output := output
       |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [], input)
  (destination, output)
      destruct Hc as [[Hno_equiv _] _].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: option message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs) l
  (finite_trace_last is tr', input)
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) :=
  VLSM.l
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |} in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i,
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Htx: (let (i, li) :=
   VLSM.l
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |} in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i,
     VLSM.input
       {|
         l := l0;
         input := input;
         destination := destination;
         output := output
       |}) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(VLSM.destination
   {|
     l := l0;
     input := input;
     destination := destination;
     output := output
   |},
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [], input)
  (destination, output)
      simpl in Htx,Hvx,Hstate_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: option message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs) l
  (finite_trace_last is tr', input)
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is tr') i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (equivocators_state_project IM
        final_descriptors'
        (finite_trace_last is tr') i, input) in
 (state_update IM
    (equivocators_state_project IM
       final_descriptors'
       (finite_trace_last is tr')) i si', om')) =
(destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [], input)
  (destination, output)
      rewrite Hstate_project in Hvx, Htx.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: option message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs) l
  (finite_trace_last is tr', input)
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i, input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i, input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [], input)
  (destination, output)
      destruct input as [input|]
      ; [|repeat split; [assumption| apply option_valid_message_None |assumption| apply HconstraintNone |assumption]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs) l
  (finite_trace_last is tr', Some input)
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      simpl in Hno_equiv.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_no_equivocations
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs) l
  (finite_trace_last is tr', Some input)
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      apply or_comm in Hno_equiv.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: False
∨ composite_has_been_sent
    (equivocator_IM IM)
    (equivocator_Hbs IM Hbs)
    (finite_trace_last is tr', Some input).1
    input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      destruct Hno_equiv as [Hinit_input | Hno_equiv]
      ; [contradiction|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      assert
        (Hs_free : valid_state_prop FreeE (finite_trace_last is tr')).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
valid_state_prop FreeE (finite_trace_last is tr')
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX') [], 
  Some input) 
  (destination, output)
      {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
valid_state_prop FreeE (finite_trace_last is tr') destruct Hs as [_om Hs].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
_om: option message
Hs: valid_state_message_prop XE
  (finite_trace_last is tr') _om
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
valid_state_prop FreeE (finite_trace_last is tr')
        apply (constraint_subsumption_valid_state_message_preservation (equivocator_IM IM))
          with (constraint2 := free_constraint (equivocator_IM IM))
          in Hs as Hs_free
          ; [|intro x; intros; exact I].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
_om: option message
Hs: valid_state_message_prop XE
  (finite_trace_last is tr') _om
Hs_free: valid_state_message_prop
  (composite_vlsm 
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)))
  (finite_trace_last is tr') _om
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
valid_state_prop FreeE (finite_trace_last is tr')
        exists _om.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
_om: option message
Hs: valid_state_message_prop XE
  (finite_trace_last is tr') _om
Hs_free: valid_state_message_prop
  (composite_vlsm 
     (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)))
  (finite_trace_last is tr') _om
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
valid_state_message_prop FreeE
  (finite_trace_last is tr') _om assumption.
      }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      specialize
        (specialized_proper_sent_rev FreeE _ Hs_free _ Hno_equiv) as Hall.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: specialized_selected_message_exists_in_all_traces
  FreeE (field_selector VLSM.output)
  (finite_trace_last is tr') input
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      specialize (Hall is tr' (valid_trace_add_default_last Htr')).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      destruct (equivocators_trace_project_output_reflecting_inv IM _ _ (proj1 Htr'pre) _ Hall)
        as [final_descriptors_m [initial_descriptors_m [trXm [_Hfinal_descriptors_m [Hproject_trXm Hex]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      assert (Hfinal_descriptors_m : proper_fixed_equivocator_descriptors final_descriptors_m (finite_trace_last is tr')).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
proper_fixed_equivocator_descriptors
  final_descriptors_m (finite_trace_last is tr')
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX') [], 
  Some input) 
  (destination, output)
      {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
proper_fixed_equivocator_descriptors
  final_descriptors_m (finite_trace_last is tr') apply not_equivocating_equivocator_descriptors_proper_fixed; [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
valid_state_prop XE (finite_trace_last is tr')
        apply finite_valid_trace_last_pstate.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
finite_valid_trace_from XE is tr' assumption.
      }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input :=
         VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
        destination := s;
        output :=
         VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      specialize (H (length tr')).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}])
→ ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      spec H.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}])
→ ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}])
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
H: ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
  finite_valid_trace XE is tr
  → length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM,
        proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
        → 
        ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX') [], 
  Some input) 
  (destination, output) {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}])
→ ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |};
      destination := s;
      output :=
        VLSM.output
          {|
            l := l0;
            input := Some input;
            destination := destination;
            output := output
          |}
    |}]) rewrite app_length.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}])
→ ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
length tr' <
length tr' +
length
  [{|
     l := l;
     input :=
       VLSM.input
         {|
           l := l0;
           input := Some input;
           destination := destination;
           output := output
         |};
     destination := s;
     output :=
       VLSM.output
         {|
           l := l0;
           input := Some input;
           destination := destination;
           output := output
         |}
   |}] simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
H: length tr' <
length
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}])
→ ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
    finite_valid_trace XE is tr
    → length tr' = length tr
      → ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
length tr' < length tr' + 1 lia. }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
H: ∀ tr : list
         (composite_transition_item
         (equivocator_IM IM)),
  finite_valid_trace XE is tr
  → length tr' = length tr
    → ∀ final_descriptors : 
         equivocator_descriptors IM,
        proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
        → 
        ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      specialize (H tr' (conj Htr Hinit) eq_refl final_descriptors_m Hfinal_descriptors_m).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
H: ∃ (trX : list (composite_transition_item IM)) 
  (initial_descriptors : 
   equivocator_descriptors IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM
      final_descriptors_m tr' =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM
        final_descriptors_m
        (finite_trace_last is tr') =
      finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX
      ∧ finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      destruct H as [trXm' [initial_descriptors_m' [Hproper_initial_m [Hproject_trXm' [Hpr_fin_tr' HtrXm]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: label
input: message
destination: state
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input :=
        VLSM.input
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |};
      destination := s;
      output :=
        VLSM.output
         {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
         |}
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input :=
      VLSM.input
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |};
    destination := s;
    output :=
      VLSM.output
        {|
         l := l0;
         input := Some input;
         destination := destination;
         output := output
        |}
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: transition l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |}) =
(s,
VLSM.output
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |})
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr', Some input).1
  input
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr',
  VLSM.input
    {|
      l := l0;
      input := Some input;
      destination := destination;
      output := output
    |})
Hiom: option_valid_message_prop XE
  (VLSM.input
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |})
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: initial_state_prop
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s =
VLSM.destination
  {|
    l := l0;
    input := Some input;
    destination := destination;
    output := output
  |}
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
trXm': list (composite_transition_item IM)
initial_descriptors_m': equivocator_descriptors IM
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m' is
Hproject_trXm': equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm', initial_descriptors_m')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      simpl in *.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
trXm': list (composite_transition_item IM)
initial_descriptors_m': equivocator_descriptors IM
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m' is
Hproject_trXm': equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm', initial_descriptors_m')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output) rewrite Hproject_trXm in Hproject_trXm'.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
trXm': list (composite_transition_item IM)
initial_descriptors_m': equivocator_descriptors IM
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m' is
Hproject_trXm': Some (trXm, initial_descriptors_m) =
Some (trXm', initial_descriptors_m')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      inversion Hproject_trXm'.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
trXm': list (composite_transition_item IM)
initial_descriptors_m': equivocator_descriptors IM
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m' is
Hproject_trXm': Some (trXm, initial_descriptors_m) =
Some (trXm', initial_descriptors_m')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m' is) trXm'
H0: trXm = trXm'
H1: initial_descriptors_m = initial_descriptors_m'
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output) subst trXm' initial_descriptors_m'.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm': Some (trXm, initial_descriptors_m) =
Some (trXm, initial_descriptors_m)
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output) clear Hproject_trXm'.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
input_valid_transition X' l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input) (destination, output)
      repeat split; [assumption| |assumption| |assumption]
      ; [ apply option_valid_message_Some
        ; apply (valid_trace_output_is_valid X' _ _ (proj1 HtrXm) _ Hex)
        |].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
constraint l0
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX') [],
  Some input)
      rewrite <- Hstate_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
constraint l0
  (finite_trace_last
     (equivocators_state_project IM final_descriptors'
        (finite_trace_last is tr')) [], Some input)
      apply Hconstraint_hbs; [assumption| apply Hproper'|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
has_been_sent FreeE (finite_trace_last is tr') input
      assert (Hlst'pre : valid_state_prop (pre_loaded_with_all_messages_vlsm FreeE) (finite_trace_last is tr')).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr')
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
Hlst'pre: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr')
has_been_sent FreeE (finite_trace_last is tr') input
      {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr') apply finite_valid_trace_last_pstate.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) is tr' apply Htr'pre. }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
Hlst'pre: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr')
has_been_sent FreeE (finite_trace_last is tr') input
      apply proper_sent; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
Hlst'pre: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr')
selected_message_exists_in_all_preloaded_traces FreeE
  (field_selector VLSM.output)
  (finite_trace_last is tr') input
      apply has_been_sent_consistency; [assumption| assumption| ].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
Hlst'pre: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr')
selected_message_exists_in_some_preloaded_traces FreeE
  (field_selector VLSM.output)
  (finite_trace_last is tr') input
      exists is, tr', (valid_trace_add_default_last Htr'pre).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
s: _composite_state (equivocator_IM IM)
l0: _composite_label IM
input: message
destination: _composite_state IM
output: option message
final_descriptors, final_descriptors': equivocator_descriptors IM
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := Some input;
      destination := s;
      output := output
    |}]): _composite_state (equivocator_IM IM)
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := Some input;
    destination := s;
    output := output
  |} =
Some
  (Some
     {|
       l := l0;
       input := Some input;
       destination := destination;
       output := output
     |}, final_descriptors')
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is tr' i, Some input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is tr') i si', om')) =
(s, output)
Hno_equiv: composite_has_been_sent
  (equivocator_IM IM)
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is tr') input
Hv: let (i, li) := l in
vvalid (equivocator_IM IM i) li
  (finite_trace_last is tr' i, Some input)
Hiom: option_valid_message_prop XE (Some input)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace_from X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
His: composite_initial_state_prop IM
  (equivocators_state_project IM
     initial_descriptors is)
Hl: ∃ Hex : existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
  existT (projT1 l)
    (existing_equivocator_label_extract
       (IM (projT1 l)) 
       (projT2 l) Hex) = l0
Hdestination: equivocators_state_project IM
  final_descriptors s = destination
Hvx: let (i, li) := l0 in
vvalid (IM i) li
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX' i,
  Some input)
Htx: (let (i, li) := l0 in
 let (si', om') :=
   vtransition 
     (IM i) li
     (finite_trace_last
        (equivocators_state_project IM
         initial_descriptors is) trX' i,
     Some input) in
 (state_update IM
    (finite_trace_last
       (equivocators_state_project IM
         initial_descriptors is) trX') i si',
 om')) = (destination, output)
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hplst: valid_state_prop X'
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX')
Hs_free: valid_state_prop FreeE
  (finite_trace_last is tr')
Hall: trace_has_message 
  (field_selector VLSM.output) input tr'
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
_Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr')
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr' =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector VLSM.output input) trXm
Hfinal_descriptors_m: proper_fixed_equivocator_descriptors
  final_descriptors_m
  (finite_trace_last is tr')
Hpr_fin_tr': equivocators_state_project IM
  final_descriptors_m
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
HtrXm: finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors_m is) trXm
Hproper_initial_m: proper_fixed_equivocator_descriptors
  initial_descriptors_m is
Hlst'pre: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is tr')
trace_has_message (field_selector VLSM.output) input
  tr' assumption.
    +message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX exists trX'.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
∃ initial_descriptors : equivocator_descriptors IM,
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ foldr (equivocators_trace_project_folder IM) foldx
      tr' = Some (trX', initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state =
      finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) trX'
      ∧ finite_valid_trace X'
          (equivocators_state_project IM
             initial_descriptors is) trX' exists initial_descriptors.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
foldx: option
  (list (composite_transition_item IM) *
   equivocator_descriptors IM)
Heqfoldx: foldx = Some ([], final_descriptors')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
proper_fixed_equivocator_descriptors
  initial_descriptors is
∧ foldr (equivocators_trace_project_folder IM) foldx
    tr' = Some (trX', initial_descriptors)
  ∧ equivocators_state_project IM final_descriptors
      final_state =
    finite_trace_last
      (equivocators_state_project IM
         initial_descriptors is) trX'
    ∧ finite_valid_trace X'
        (equivocators_state_project IM
           initial_descriptors is) trX' subst foldx.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
proper_fixed_equivocator_descriptors
  initial_descriptors is
∧ foldr (equivocators_trace_project_folder IM)
    (Some ([], final_descriptors')) tr' =
  Some (trX', initial_descriptors)
  ∧ equivocators_state_project IM final_descriptors
      final_state =
    finite_trace_last
      (equivocators_state_project IM
         initial_descriptors is) trX'
    ∧ finite_valid_trace X'
        (equivocators_state_project IM
           initial_descriptors is) trX' split; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
foldr (equivocators_trace_project_folder IM)
  (Some ([], final_descriptors')) tr' =
Some (trX', initial_descriptors)
∧ equivocators_state_project IM final_descriptors
    final_state =
  finite_trace_last
    (equivocators_state_project IM initial_descriptors
       is) trX'
  ∧ finite_valid_trace X'
      (equivocators_state_project IM
         initial_descriptors is) trX'
      split; [apply Htr_project|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM initial_descriptors
     is) trX'
∧ finite_valid_trace X'
    (equivocators_state_project IM initial_descriptors
       is) trX' split; [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr, tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is tr: _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
Htr_lst: tr =
tr' ++
[{|
   l := l;
   input := iom;
   destination := s;
   output := oom
 |}]
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM initial_descriptors
     is) trX'
      subst tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
constraint: composite_label IM
→ composite_state IM * option message
  → Prop
HconstraintNone: ∀ (l : composite_label IM) 
  (s : composite_state IM),
  constraint l (s, None)
Hconstraint_hbs: constraint_has_been_sent_prop
  constraint
X':= composite_vlsm IM constraint: VLSM message
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
H: ∀ y : nat,
  y <
  length
    (tr' ++
     [{|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |}])
  → ∀ tr : 
       list
         (composite_transition_item
         (equivocator_IM IM)),
      finite_valid_trace XE is tr
      → y = length tr
        → 
        ∀ final_descriptors : 
         equivocator_descriptors IM,
         proper_fixed_equivocator_descriptors
         final_descriptors
         (finite_trace_last is tr)
         → 
         ∃ (trX : 
         list 
         (composite_transition_item IM)) 
         (initial_descriptors : 
         equivocator_descriptors IM),
         proper_fixed_equivocator_descriptors
         initial_descriptors is
         ∧ 
         equivocators_trace_project IM
         final_descriptors tr =
         Some (trX, initial_descriptors)
         ∧ 
         equivocators_state_project IM
         final_descriptors
         (finite_trace_last is tr) =
         finite_trace_last
         (equivocators_state_project IM
         initial_descriptors is) trX
         ∧ 
         finite_valid_trace X'
         (equivocators_state_project IM
         initial_descriptors is) trX
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
Htr: finite_valid_trace_from XE is tr'
Hs: valid_state_prop XE (finite_trace_last is tr')
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is tr', iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is tr', iom)
Ht: transition l (finite_trace_last is tr', iom) =
(s, oom)
Hinit: composite_initial_state_prop
  (equivocator_IM IM) is
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors s
final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors' tr' =
Some (trX', initial_descriptors)
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
HtrX': finite_valid_trace X'
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hprojectx: equivocators_transition_item_project IM
  final_descriptors
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some (None, final_descriptors')
Hitemx: True
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
Hproper': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is tr')
Htr': finite_valid_trace FreeE is tr'
Htr'pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr'
Hproper_initial: proper_fixed_equivocator_descriptors
  initial_descriptors is
equivocators_state_project IM final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM initial_descriptors
     is) trX' clear -Hstate_project Hx.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
is: composite_state (equivocator_IM IM)
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
final_descriptors, final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hx: equivocators_state_project IM final_descriptors s =
equivocators_state_project IM final_descriptors'
  (finite_trace_last is tr')
equivocators_state_project IM final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM initial_descriptors
     is) trX'
      rewrite Hstate_project in Hx.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
is: composite_state (equivocator_IM IM)
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
final_descriptors, final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hx: equivocators_state_project IM final_descriptors s =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
equivocators_state_project IM final_descriptors
  final_state =
finite_trace_last
  (equivocators_state_project IM initial_descriptors
     is) trX'
      rewrite <- Hx.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
is: composite_state (equivocator_IM IM)
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
final_descriptors, final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hx: equivocators_state_project IM final_descriptors s =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
equivocators_state_project IM final_descriptors
  final_state =
equivocators_state_project IM final_descriptors s f_equal.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
is: composite_state (equivocator_IM IM)
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
final_descriptors, final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hx: equivocators_state_project IM final_descriptors s =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
final_state = s unfold final_state.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
is: composite_state (equivocator_IM IM)
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
final_descriptors, final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hx: equivocators_state_project IM final_descriptors s =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]) = s
      rewrite finite_trace_last_is_last.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
is: composite_state (equivocator_IM IM)
tr': list
  (composite_transition_item (equivocator_IM IM))
l: _composite_label (equivocator_IM IM)
iom: option message
s: _composite_state (equivocator_IM IM)
oom: option message
final_state:= finite_trace_last is
  (tr' ++
   [{|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}]): _composite_state (equivocator_IM IM)
final_descriptors, final_descriptors': equivocator_descriptors IM
trX': list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hstate_project: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is tr') =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
Hx: equivocators_state_project IM final_descriptors s =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX'
destination
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = s
      reflexivity.
Qed.

Instantiating the lemma above with the free constraint.

Lemma free_equivocators_valid_trace_project
  (final_descriptors : equivocator_descriptors IM)
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (final_state := @finite_trace_last _ (@type _ XE) is tr)
  (Hproper : proper_fixed_equivocator_descriptors final_descriptors final_state)
  (Htr : finite_valid_trace XE is tr)
  : exists
    (trX : list (composite_transition_item IM))
    (initial_descriptors : equivocator_descriptors IM)
    (isX := equivocators_state_project IM initial_descriptors is)
    (final_stateX := finite_trace_last isX trX),
    proper_fixed_equivocator_descriptors initial_descriptors is /\
    equivocators_trace_project IM final_descriptors tr = Some (trX, initial_descriptors) /\
    equivocators_state_project IM final_descriptors final_state = final_stateX /\
    finite_valid_trace (free_composite_vlsm IM) isX trX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace (free_composite_vlsm IM)
          isX trX
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace (free_composite_vlsm IM)
          isX trX
  apply _equivocators_valid_trace_project; [assumption | assumption| ..]
  ; unfold constraint_has_been_sent_prop; intros; exact I.
Qed.

A message sent by a non-state-equivocating machine can be observed in any projection of the final state.

Lemma not_equivocating_sent_message_has_been_observed_in_projection
  (is: vstate XE)
  (tr: list (composite_transition_item (equivocator_IM IM)))
  (Htr: finite_valid_trace XE is tr)
  (lst := finite_trace_last is tr)
  (item: transition_item)
  (Hitem: item ∈ tr)
  (Hitem_not_equiv: (projT1 (l item)) ∉ equivocating)
  (m: message)
  (Hm: field_selector output m item)
  (descriptors: equivocator_descriptors IM)
  (Hdescriptors: proper_fixed_equivocator_descriptors descriptors lst)
  : has_been_observed Free (equivocators_state_project IM descriptors lst) m.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
lst:= finite_trace_last is tr: state
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors lst
has_been_observed Free
  (equivocators_state_project IM descriptors lst) m
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
lst:= finite_trace_last is tr: state
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors lst
has_been_observed Free
  (equivocators_state_project IM descriptors lst) m
  destruct (free_equivocators_valid_trace_project descriptors is tr Hdescriptors Htr)
    as [trX [initial_descriptors [_ [Htr_project [Hfinal_state HtrX_Free]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
lst:= finite_trace_last is tr: state
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors lst
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
has_been_observed Free
  (equivocators_state_project IM descriptors lst) m
  subst lst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
has_been_observed Free
  (equivocators_state_project IM descriptors
     (finite_trace_last is tr)) m
  rewrite Hfinal_state.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
has_been_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m

  assert (Htr_Pre : finite_valid_trace (pre_loaded_with_all_messages_vlsm FreeE) is tr).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE) is tr
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
has_been_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE) is tr revert Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_valid_trace XE is tr
→ finite_valid_trace
    (pre_loaded_with_all_messages_vlsm FreeE) is tr apply VLSM_incl_finite_valid_trace.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
VLSM_incl_part
  (composite_vlsm_machine (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint IM
        Hbs index_listing equivocating))
  (pre_loaded_with_all_messages_vlsm_machine FreeE)
    apply VLSM_incl_trans with (machine FreeE);
    [|apply vlsm_incl_pre_loaded_with_all_messages_vlsm].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
VLSM_incl
  (mk_vlsm
     {|
       transition :=
         composite_transition (equivocator_IM IM);
       valid :=
         constrained_composite_valid
           (equivocator_IM IM)
           (equivocators_fixed_equivocations_constraint
              IM Hbs index_listing equivocating)
     |}) (mk_vlsm (machine FreeE))
    apply equivocators_fixed_equivocations_vlsm_incl_free.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
has_been_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m

  assert (HtrX_Pre : finite_valid_trace (pre_loaded_with_all_messages_vlsm Free) (equivocators_state_project IM initial_descriptors is) trX ).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM initial_descriptors
     is) trX
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
has_been_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM initial_descriptors
     is) trX revert HtrX_Free.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM initial_descriptors
     is) trX
→ finite_valid_trace
    (pre_loaded_with_all_messages_vlsm Free)
    (equivocators_state_project IM initial_descriptors
       is) trX apply VLSM_incl_finite_valid_trace.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
VLSM_incl_part
  (composite_vlsm_machine IM (free_constraint IM))
  (pre_loaded_with_all_messages_vlsm_machine Free) apply vlsm_incl_pre_loaded_with_all_messages_vlsm. }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
has_been_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m

  assert (Hlst_preX : valid_state_prop (pre_loaded_with_all_messages_vlsm Free) (finite_trace_last (equivocators_state_project IM initial_descriptors is) trX)).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
has_been_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) apply (finite_valid_trace_last_pstate (pre_loaded_with_all_messages_vlsm Free)).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM initial_descriptors
     is) trX
    apply HtrX_Pre.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
has_been_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m

  rewrite has_been_observed_sent_received_iff; [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
has_been_received Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
∨ has_been_sent Free
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) trX) m
  right.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
has_been_sent Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m apply proper_sent; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
selected_message_exists_in_all_preloaded_traces Free
  (field_selector output)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  apply has_been_sent_consistency; [assumption| assumption |].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
selected_message_exists_in_some_preloaded_traces Free
  (field_selector output)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  exists (equivocators_state_project IM initial_descriptors is), trX,
    (valid_trace_add_default_last HtrX_Pre).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
trace_has_message (field_selector output) m trX

  apply elem_of_list_split in Hitem.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: ∃ l1 l2 : list transition_item,
  tr = l1 ++ item :: l2
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
trace_has_message (field_selector output) m trX
  destruct Hitem as [pre [suf Hitem]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
pre, suf: list transition_item
Hitem: tr = pre ++ item :: suf
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
trace_has_message (field_selector output) m trX
  change (item :: suf) with ([item] ++ suf) in Hitem.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
pre, suf: list transition_item
Hitem: tr = pre ++ [item] ++ suf
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
trace_has_message (field_selector output) m trX
  subst tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
trace_has_message (field_selector output) m trX

  assert (Hsingleton_d_item : is_singleton_state (IM (projT1 (VLSM.l item))) (destination item (projT1 (VLSM.l item)))).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
is_singleton_state (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
trace_has_message (field_selector output) m trX
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
is_singleton_state (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
    apply (not_equivocating_index_has_singleton_state IM Hbs _ finite_index equivocating); [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM Hbs
     index_listing equivocating) (destination item)
    apply proj1 in Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace_from XE is
  (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM Hbs
     index_listing equivocating) (destination item)
    rewrite app_assoc in Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM Hbs
     index_listing equivocating) (destination item)
    apply finite_valid_trace_from_app_iff in Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace_from XE is (pre ++ [item])
∧ finite_valid_trace_from XE
    (finite_trace_last is (pre ++ [item])) suf
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM Hbs
     index_listing equivocating) (destination item)
    destruct Htr as [Htr _].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace_from XE is (pre ++ [item])
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM Hbs
     index_listing equivocating) (destination item)
    apply finite_valid_trace_last_pstate in Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: valid_state_prop XE
  (finite_trace_last is (pre ++ [item]))
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM Hbs
     index_listing equivocating) (destination item)
    rewrite finite_trace_last_is_last in Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: valid_state_prop XE (destination item)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM Hbs
     index_listing equivocating) (destination item) assumption.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
trace_has_message (field_selector output) m trX

  apply equivocators_trace_project_app_iff in Htr_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: ∃ (preX
   sufX : 
   list (composite_transition_item IM)) 
  (eqv_descriptors' : 
   equivocator_descriptors IM),
  equivocators_trace_project IM
    descriptors 
    ([item] ++ suf) =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project IM
      eqv_descriptors' pre =
    Some (preX, initial_descriptors)
    ∧ trX = preX ++ sufX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
trace_has_message (field_selector output) m trX
  destruct Htr_project as [preX' [sufX' [eqv_descriptors' [Hpr_suf [Hpr_pre HpreX]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
preX', sufX': list (composite_transition_item IM)
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HpreX: trX = preX' ++ sufX'
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
trace_has_message (field_selector output) m trX
  subst trX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
trace_has_message (field_selector output) m
  (preX' ++ sufX') apply Exists_app.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) preX'
∨ Exists (field_selector output m) sufX' right.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) sufX'

  apply proj1 in Htr_Pre.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Htr_Pre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) sufX'
  apply finite_valid_trace_from_app_iff in Htr_Pre.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Htr_Pre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  is pre
∧ finite_valid_trace_from
    (pre_loaded_with_all_messages_vlsm FreeE)
    (finite_trace_last is pre)
    ([item] ++ suf)
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) sufX'
  destruct Htr_Pre as [Hpre'_free Hsuf'_free].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM descriptors
  ([item] ++ suf) =
Some (sufX', eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) 
  (finite_trace_last is pre)
  ([item] ++ suf)
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) sufX'

  apply equivocators_trace_project_app_iff in Hpr_suf.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
Hpr_suf: ∃ (preX
   sufX : 
   list (composite_transition_item IM)) 
  (eqv_descriptors'0 : 
   equivocator_descriptors IM),
  equivocators_trace_project IM descriptors
    suf = 
  Some (sufX, eqv_descriptors'0)
  ∧ equivocators_trace_project IM
      eqv_descriptors'0 [item] =
    Some (preX, eqv_descriptors')
    ∧ sufX' = preX ++ sufX
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) 
  (finite_trace_last is pre)
  ([item] ++ suf)
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) sufX'
  destruct Hpr_suf as [pre_itemX [sufX'' [eqv_descriptors'' [Hpr_suf' [Hpr_pre_item HsufX']]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
eqv_descriptors': equivocator_descriptors IM
pre_itemX, sufX'': list (composite_transition_item IM)
eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
HsufX': sufX' = pre_itemX ++ sufX''
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) 
  (finite_trace_last is pre)
  ([item] ++ suf)
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ sufX'))
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ sufX')
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) sufX'
  subst sufX'.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) 
  (finite_trace_last is pre)
  ([item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) (pre_itemX ++ sufX'') apply Exists_app.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) 
  (finite_trace_last is pre)
  ([item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) pre_itemX
∨ Exists (field_selector output m) sufX'' left.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) 
  (finite_trace_last is pre)
  ([item] ++ suf)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) pre_itemX

  apply finite_valid_trace_from_app_iff in Hsuf'_free.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) 
  (finite_trace_last is pre) [item]
∧ finite_valid_trace_from
    (pre_loaded_with_all_messages_vlsm
       FreeE)
    (finite_trace_last
       (finite_trace_last is pre) [item])
    suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) pre_itemX
  destruct Hsuf'_free as [Hpre_item_free Hsuf'_free].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Exists (field_selector output m) pre_itemX
  assert (Heqv_descriptors'' : forall i, i ∉ equivocating -> eqv_descriptors'' i = Existing 0).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
∀ i : index,
  i ∉ equivocating → eqv_descriptors'' i = Existing 0
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Heqv_descriptors'': ∀ i : index,
  i ∉ equivocating
  → eqv_descriptors'' i =
    Existing 0
Exists (field_selector output m) pre_itemX
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
∀ i : index,
  i ∉ equivocating → eqv_descriptors'' i = Existing 0 specialize (equivocators_trace_project_preserves_zero_descriptors IM _ _ Hsuf'_free descriptors) as Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Hpr: ∀ (idescriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project IM descriptors suf =
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
∀ i : index,
  i ∉ equivocating → eqv_descriptors'' i = Existing 0
    specialize (Hpr _ _ Hpr_suf').message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Hpr: ∀ i : index,
  descriptors i = Existing 0
  → eqv_descriptors'' i = Existing 0
∀ i : index,
  i ∉ equivocating → eqv_descriptors'' i = Existing 0
    intros i Hi.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Hpr: ∀ i : index,
  descriptors i = Existing 0
  → eqv_descriptors'' i = Existing 0
i: index
Hi: i ∉ equivocating
eqv_descriptors'' i = Existing 0
    apply Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Hpr: ∀ i : index,
  descriptors i = Existing 0
  → eqv_descriptors'' i = Existing 0
i: index
Hi: i ∉ equivocating
descriptors i = Existing 0 apply Hdescriptors.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Hpr: ∀ i : index,
  descriptors i = Existing 0
  → eqv_descriptors'' i = Existing 0
i: index
Hi: i ∉ equivocating
i ∉ equivocating assumption.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Heqv_descriptors'': ∀ i : index,
  i ∉ equivocating
  → eqv_descriptors'' i =
    Existing 0
Exists (field_selector output m) pre_itemX

  specialize (Heqv_descriptors'' _ Hitem_not_equiv).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: equivocators_trace_project IM
  eqv_descriptors'' [item] =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Heqv_descriptors'': eqv_descriptors''
  (projT1 (l item)) = 
Existing 0
Exists (field_selector output m) pre_itemX
  simpl in *.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ item :: suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: output item = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ item :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ item :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: match
  equivocators_transition_item_project
    IM eqv_descriptors'' item
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Heqv_descriptors'': eqv_descriptors''
  (projT1 (l item)) = 
Existing 0
Exists (field_selector output m) pre_itemX
  destruct (equivocators_transition_item_project IM eqv_descriptors'' item)
    as [(o, odescriptor)|] eqn:Hpr
  ; [|congruence].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
item: transition_item
pre, suf: list transition_item
Htr: finite_valid_trace XE is (pre ++ item :: suf)
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: output item = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ item :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ item :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors'' item = 
Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre) [item]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre) [item])
  suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (l item)))
  (destination item
     (projT1 (l item)))
Heqv_descriptors'': eqv_descriptors''
  (projT1 (l item)) = 
Existing 0
Exists (field_selector output m) pre_itemX
  destruct item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
l: label
input: option message
destination: state
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: projT1
  (VLSM.l
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |}) ∉ equivocating
m: message
Hm: VLSM.output
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |} = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre)
  [{|
     l := l;
     input := input;
     destination := destination;
     output := output
   |}]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := l;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM
     (projT1
        (VLSM.l
         {|
         l := l;
         input := input;
         destination :=
         destination;
         output := output
         |})))
  (VLSM.destination
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |}
     (projT1
        (VLSM.l
         {|
         l := l;
         input := input;
         destination :=
         destination;
         output := output
         |})))
Heqv_descriptors'': eqv_descriptors''
  (projT1
     (VLSM.l
        {|
         l := l;
         input := input;
         destination :=
         destination;
         output := output
        |})) = 
Existing 0
Exists (field_selector VLSM.output m) pre_itemX simpl in *.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
l: _composite_label (equivocator_IM IM)
input: option message
destination: _composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: projT1 l ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last is pre)
  [{|
     l := l;
     input := input;
     destination := destination;
     output := output
   |}]
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := l;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state 
  (IM (projT1 l))
  (destination (projT1 l))
Heqv_descriptors'': eqv_descriptors'' (projT1 l) =
Existing 0
Exists (field_selector VLSM.output m) pre_itemX
  apply first_transition_valid in Hpre_item_free.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
l: _composite_label (equivocator_IM IM)
input: option message
destination: _composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: projT1 l ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: input_valid_transition
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (VLSM.l
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |})
  (finite_trace_last is pre,
  VLSM.input
    {|
      l := l;
      input := input;
      destination := destination;
      output := output
    |})
  (VLSM.destination
     {|
       l := l;
       input := input;
       destination := destination;
       output := output
     |},
  VLSM.output
    {|
      l := l;
      input := input;
      destination := destination;
      output := output
    |})
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := l;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state 
  (IM (projT1 l))
  (destination (projT1 l))
Heqv_descriptors'': eqv_descriptors'' (projT1 l) =
Existing 0
Exists (field_selector VLSM.output m) pre_itemX simpl in Hpre_item_free.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
l: _composite_label (equivocator_IM IM)
input: option message
destination: _composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: projT1 l ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hpre_item_free: input_valid_transition
  (pre_loaded_with_all_messages_vlsm
     FreeE) l
  (finite_trace_last is pre, input)
  (destination, output)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := l;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state 
  (IM (projT1 l))
  (destination (projT1 l))
Heqv_descriptors'': eqv_descriptors'' (projT1 l) =
Existing 0
Exists (field_selector VLSM.output m) pre_itemX
  destruct Hpre_item_free as [[_ [_ [Hv _]]] Ht].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
l: _composite_label (equivocator_IM IM)
input: option message
destination: _composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := l;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: projT1 l ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := l;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is pre, input)
Ht: transition l (finite_trace_last is pre, input) =
(destination, output)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := l;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state 
  (IM (projT1 l))
  (destination (projT1 l))
Heqv_descriptors'': eqv_descriptors'' (projT1 l) =
Existing 0
Exists (field_selector VLSM.output m) pre_itemX
  destruct l.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
v: vlabel (equivocator_IM IM x)
input: option message
destination: _composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x v;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: projT1 (existT x v) ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x v;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hv: composite_valid 
  (equivocator_IM IM) 
  (existT x v) 
  (finite_trace_last is pre, input)
Ht: transition 
  (existT x v) 
  (finite_trace_last is pre, input) =
(destination, output)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state
  (IM (projT1 (existT x v)))
  (destination
     (projT1 (existT x v)))
Heqv_descriptors'': eqv_descriptors''
  (projT1 (existT x v)) =
Existing 0
Exists (field_selector VLSM.output m) pre_itemX simpl in *.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
v: vlabel (equivocator_IM IM x)
input: option message
destination: _composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x v;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x v;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hv: vvalid (equivocator_IM IM x) v
  (finite_trace_last is pre x, input)
Ht: vtransition FreeE 
  (existT x v) 
  (finite_trace_last is pre, input) =
(destination, output)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state 
  (IM x) (destination x)
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector VLSM.output m) pre_itemX unfold vtransition in Ht.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
v: vlabel (equivocator_IM IM x)
input: option message
destination: _composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x v;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x v;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hv: vvalid (equivocator_IM IM x) v
  (finite_trace_last is pre x, input)
Ht: transition 
  (existT x v) 
  (finite_trace_last is pre, input) =
(destination, output)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state 
  (IM x) (destination x)
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector VLSM.output m) pre_itemX simpl in Ht.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
v: vlabel (equivocator_IM IM x)
input: option message
destination: _composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x v;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x v;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hv: vvalid (equivocator_IM IM x) v
  (finite_trace_last is pre x, input)
Ht: (let (si', om') :=
   vtransition 
     (equivocator_IM IM x) v
     (finite_trace_last is pre x, input) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is pre) x si', om')) =
(destination, output)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state 
  (IM x) (destination x)
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector VLSM.output m) pre_itemX
  match type of Ht with
  | (let (_, _) := ?t in _) = _ => destruct t as (si', om') eqn:Hti
  end.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
v: vlabel (equivocator_IM IM x)
input: option message
destination: _composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x v;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x v;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hv: vvalid (equivocator_IM IM x) v
  (finite_trace_last is pre x, input)
si': state
om': option message
Hti: vtransition 
  (equivocator_IM IM x) v
  (finite_trace_last is pre x, input) =
(si', om')
Ht: (state_update 
   (equivocator_IM IM) 
   (finite_trace_last is pre) x si', om') =
(destination, output)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state 
  (IM x) (destination x)
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector VLSM.output m) pre_itemX
  inversion Ht.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
v: vlabel (equivocator_IM IM x)
input: option message
destination: _composite_state (equivocator_IM IM)
output: option message
pre, suf: list transition_item
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x v;
     input := input;
     destination := destination;
     output := output
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
m: message
Hm: output = Some m
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x v;
    input := input;
    destination := destination;
    output := output
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hv: vvalid (equivocator_IM IM x) v
  (finite_trace_last is pre x, input)
si': state
om': option message
Hti: vtransition 
  (equivocator_IM IM x) v
  (finite_trace_last is pre x, input) =
(si', om')
Ht: (state_update 
   (equivocator_IM IM) 
   (finite_trace_last is pre) x si', om') =
(destination, output)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x v;
        input := input;
        destination := destination;
        output := output
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state 
  (IM x) (destination x)
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
H0: state_update 
  (equivocator_IM IM) 
  (finite_trace_last is pre) x si' = destination
H1: om' = output
Exists (field_selector VLSM.output m) pre_itemX subst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
v: vlabel (equivocator_IM IM x)
input: option message
pre, suf: list transition_item
m: message
si': state
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x v;
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x v;
    input := input;
    destination :=
      state_update 
        (equivocator_IM IM)
        (finite_trace_last is pre) x si';
    output := Some m
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hv: vvalid (equivocator_IM IM x) v
  (finite_trace_last is pre x, input)
Ht: (state_update 
   (equivocator_IM IM) 
   (finite_trace_last is pre) x si', 
Some m) =
(state_update 
   (equivocator_IM IM) 
   (finite_trace_last is pre) x si', 
Some m)
Hti: vtransition 
  (equivocator_IM IM x) v
  (finite_trace_last is pre x, input) =
(si', Some m)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x v;
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state 
  (IM x)
  (state_update
     (equivocator_IM IM)
     (finite_trace_last is pre) x
     si' x)
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX rewrite state_update_eq in Hsingleton_d_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
v: vlabel (equivocator_IM IM x)
input: option message
pre, suf: list transition_item
m: message
si': state
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x v;
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x v;
    input := input;
    destination :=
      state_update 
        (equivocator_IM IM)
        (finite_trace_last is pre) x si';
    output := Some m
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hv: vvalid (equivocator_IM IM x) v
  (finite_trace_last is pre x, input)
Ht: (state_update 
   (equivocator_IM IM) 
   (finite_trace_last is pre) x si', 
Some m) =
(state_update 
   (equivocator_IM IM) 
   (finite_trace_last is pre) x si', 
Some m)
Hti: vtransition 
  (equivocator_IM IM x) v
  (finite_trace_last is pre x, input) =
(si', Some m)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x v;
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX clear Ht.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
v: vlabel (equivocator_IM IM x)
input: option message
pre, suf: list transition_item
m: message
si': state
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x v;
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x v;
    input := input;
    destination :=
      state_update 
        (equivocator_IM IM)
        (finite_trace_last is pre) x si';
    output := Some m
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hv: vvalid (equivocator_IM IM x) v
  (finite_trace_last is pre x, input)
Hti: vtransition 
  (equivocator_IM IM x) v
  (finite_trace_last is pre x, input) =
(si', Some m)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x v;
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  specialize (equivocator_transition_no_equivocation_zero_descriptor (IM x) _ _ _ _ _ Hv Hti Hsingleton_d_item)
    as [li Hsndv].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
v: vlabel (equivocator_IM IM x)
input: option message
pre, suf: list transition_item
m: message
si': state
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x v;
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: equivocators_transition_item_project IM
  eqv_descriptors''
  {|
    l := existT x v;
    input := input;
    destination :=
      state_update 
        (equivocator_IM IM)
        (finite_trace_last is pre) x si';
    output := Some m
  |} = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hv: vvalid (equivocator_IM IM x) v
  (finite_trace_last is pre x, input)
Hti: vtransition 
  (equivocator_IM IM x) v
  (finite_trace_last is pre x, input) =
(si', Some m)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x v;
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
li: vlabel (IM x)
Hsndv: v = ContinueWith 0 li
Exists (field_selector output m) pre_itemX
  unfold equivocators_transition_item_project in Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
v: vlabel (equivocator_IM IM x)
input: option message
pre, suf: list transition_item
m: message
si': state
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x v;
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM
       (projT1
         (l
         {|
         l := existT x v;
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |})))
    (composite_transition_item_projection
       (equivocator_IM IM)
       {|
         l := existT x v;
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
       |})
    (eqv_descriptors''
       (projT1
         (l
         {|
         l := existT x v;
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |})))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
         l :=
         existT
         (projT1
         (l
         {|
         l := existT x v;
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |})) 
         (l item');
         input :=
         VLSM.input
         {|
         l := existT x v;
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |};
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (destination
         {|
         l := existT x v;
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |});
         output :=
         output
         {|
         l := existT x v;
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |}
         |},
      equivocator_descriptors_update IM
        eqv_descriptors''
        (projT1
         (l
         {|
         l := existT x v;
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |})) deqv')
| Some (None, deqv') =>
    Some
      (None,
      equivocator_descriptors_update IM
        eqv_descriptors''
        (projT1
         (l
         {|
         l := existT x v;
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |})) deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hv: vvalid (equivocator_IM IM x) v
  (finite_trace_last is pre x, input)
Hti: vtransition 
  (equivocator_IM IM x) v
  (finite_trace_last is pre x, input) =
(si', Some m)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x v;
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
li: vlabel (IM x)
Hsndv: v = ContinueWith 0 li
Exists (field_selector output m) pre_itemX
  simpl in Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
v: vlabel (equivocator_IM IM x)
input: option message
pre, suf: list transition_item
m: message
si': state
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x v;
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := existT x v;
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    (composite_transition_item_projection
       (equivocator_IM IM)
       {|
         l := existT x v;
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
       |}) 
    (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
         l := existT x (l item');
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
         |},
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hv: vvalid (equivocator_IM IM x) v
  (finite_trace_last is pre x, input)
Hti: vtransition 
  (equivocator_IM IM x) v
  (finite_trace_last is pre x, input) =
(si', Some m)
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x v;
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
li: vlabel (IM x)
Hsndv: v = ContinueWith 0 li
Exists (field_selector output m) pre_itemX
  subst v.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    (composite_transition_item_projection
       (equivocator_IM IM)
       {|
         l := existT x (ContinueWith 0 li);
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
       |}) 
    (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
         l := existT x (l item');
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
         |},
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  unfold ProjectionTraces.composite_transition_item_projection in Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    (composite_transition_item_projection_from_eq
       (equivocator_IM IM)
       {|
         l := existT x (ContinueWith 0 li);
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
       |}
       (projT1
         (l
         {|
         l := existT x (ContinueWith 0 li);
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |})) eq_refl) 
    (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
         l := existT x (l item');
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
         |},
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  unfold ProjectionTraces.composite_transition_item_projection_from_eq in Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    {|
      l :=
        eq_rect_r
         (λ n : index,
         vlabel (equivocator_IM IM n))
         (projT2
         (l
         {|
         l := existT x (ContinueWith 0 li);
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |})) eq_refl;
      input :=
        VLSM.input
         {|
         l := existT x (ContinueWith 0 li);
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |};
      destination :=
        destination
         {|
         l := existT x (ContinueWith 0 li);
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |}
         (projT1
         (l
         {|
         l := existT x (ContinueWith 0 li);
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |}));
      output :=
        output
         {|
         l := existT x (ContinueWith 0 li);
         input := input;
         destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
         output := Some m
         |}
    |} (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
         l := existT x (l item');
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
         |},
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  simpl in Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    {|
      l :=
        eq_rect_r
         (λ n : index,
         vlabel (equivocator_IM IM n))
         (ContinueWith 0 li) eq_refl;
      input := input;
      destination :=
        state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si' x;
      output := Some m
    |} (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
         l := existT x (l item');
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
         |},
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  unfold eq_rect_r in Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    {|
      l :=
        eq_rect x
         (λ y : index,
         vlabel (equivocator_IM IM y))
         (ContinueWith 0 li) x 
         (eq_sym eq_refl);
      input := input;
      destination :=
        state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si' x;
      output := Some m
    |} (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
         l := existT x (l item');
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
         |},
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX simpl in Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    {|
      l := ContinueWith 0 li;
      input := input;
      destination :=
        state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si' x;
      output := Some m
    |} (eqv_descriptors'' x)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
         l := existT x (l item');
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
         |},
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  rewrite Heqv_descriptors'' in Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  equivocator_vlsm_transition_item_project
    (IM x)
    {|
      l := ContinueWith 0 li;
      input := input;
      destination :=
        state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si' x;
      output := Some m
    |} (Existing 0)
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
         l := existT x (l item');
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
         |},
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  unfold equivocator_vlsm_transition_item_project in Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  match
    equivocator_state_project
      (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si' x) 0
  with
  | Some sj =>
      if decide (0 = 0)
      then
       Some
         (
         Some
         {|
         l := li;
         input := input;
         destination := sj;
         output := Some m
         |}, 
         Existing 0)
      else Some (None, Existing 0)
  | None => None
  end
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
         l := existT x (l item');
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
         |},
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  rewrite state_update_eq in Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  match equivocator_state_project si' 0 with
  | Some sj =>
      if decide (0 = 0)
      then
       Some
         (
         Some
         {|
         l := li;
         input := input;
         destination := sj;
         output := Some m
         |}, 
         Existing 0)
      else Some (None, Existing 0)
  | None => None
  end
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
         l := existT x (l item');
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
         |},
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  rewrite equivocator_state_project_zero in Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: match
  (if decide (0 = 0)
   then
    Some
      (Some
         {|
         l := li;
         input := input;
         destination :=
         equivocator_state_zero si';
         output := Some m
         |}, 
      Existing 0)
   else Some (None, Existing 0))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
         l := existT x (l item');
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
         |},
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| Some (None, deqv') =>
    Some
      (None,
      equivocator_descriptors_update IM
        eqv_descriptors'' x deqv')
| None => None
end = Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  rewrite decide_True in Hpr by reflexivity.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: Some
  (Some
     {|
       l :=
         existT x
         (l
         {|
         l := li;
         input := input;
         destination :=
         equivocator_state_zero si';
         output := Some m
         |});
       input := input;
       destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
       output := Some m
     |},
  equivocator_descriptors_update IM
    eqv_descriptors'' x 
    (Existing 0)) = 
Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  inversion Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
o: option (composite_transition_item IM)
odescriptor: equivocator_descriptors IM
Hpr: Some
  (Some
     {|
       l :=
         existT x
         (l
         {|
         l := li;
         input := input;
         destination :=
         equivocator_state_zero si';
         output := Some m
         |});
       input := input;
       destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
       output := Some m
     |},
  equivocator_descriptors_update IM
    eqv_descriptors'' x 
    (Existing 0)) = 
Some (o, odescriptor)
Hpr_pre_item: match o with
| Some item' =>
    Some ([item'], odescriptor)
| None => Some ([], odescriptor)
end =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
H0: Some
  {|
    l := existT x li;
    input := input;
    destination :=
      equivocators_state_project IM
        eqv_descriptors''
        (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
    output := Some m
  |} = o
H1: equivocator_descriptors_update IM
  eqv_descriptors'' x 
  (Existing 0) = odescriptor
Exists (field_selector output m) pre_itemX subst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr: Some
  (Some
     {|
       l :=
         existT x
         (l
         {|
         l := li;
         input := input;
         destination :=
         equivocator_state_zero si';
         output := Some m
         |});
       input := input;
       destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
       output := Some m
     |},
  equivocator_descriptors_update IM
    eqv_descriptors'' x 
    (Existing 0)) =
Some
  (Some
     {|
       l := existT x li;
       input := input;
       destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
       output := Some m
     |},
  equivocator_descriptors_update IM
    eqv_descriptors'' x 
    (Existing 0))
Hpr_pre_item: Some
  ([{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si');
      output := Some m
    |}],
  equivocator_descriptors_update IM
    eqv_descriptors'' x 
    (Existing 0)) =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX clear Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: Some
  ([{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si');
      output := Some m
    |}],
  equivocator_descriptors_update IM
    eqv_descriptors'' x 
    (Existing 0)) =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m) pre_itemX
  inversion Hpr_pre_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', pre_itemX, sufX'': list (composite_transition_item IM)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
eqv_descriptors', eqv_descriptors'': equivocator_descriptors IM
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre_item: Some
  ([{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si');
      output := Some m
    |}],
  equivocator_descriptors_update IM
    eqv_descriptors'' x 
    (Existing 0)) =
Some (pre_itemX, eqv_descriptors')
Hpr_pre: equivocators_trace_project IM
  eqv_descriptors' pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++ pre_itemX ++ sufX'')
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++ pre_itemX ++ sufX''))
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
H0: [{|
   l := existT x li;
   input := input;
   destination :=
     equivocators_state_project IM
       eqv_descriptors''
       (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
   output := Some m
 |}] = pre_itemX
H1: equivocator_descriptors_update IM
  eqv_descriptors'' x 
  (Existing 0) = eqv_descriptors'
Exists (field_selector output m)
  [{|
     l := existT x li;
     input := input;
     destination :=
       equivocators_state_project IM eqv_descriptors''
         (state_update (equivocator_IM IM)
            (finite_trace_last is pre) x si');
     output := Some m
   |}] subst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX'': list (composite_transition_item IM)
eqv_descriptors'': equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si');
      output := Some m
    |}] ++ sufX'')
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre: equivocators_trace_project IM
  (equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0)) pre =
Some (preX', initial_descriptors)
Hpr_pre_item: Some
  ([{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si');
      output := Some m
    |}],
  equivocator_descriptors_update IM
    eqv_descriptors'' x 
    (Existing 0)) =
Some
  ([{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si');
      output := Some m
    |}],
  equivocator_descriptors_update IM
    eqv_descriptors'' x 
    (Existing 0))
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
      output := Some m
    |}] ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++
      [{|
         l := existT x li;
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
       |}] ++ sufX''))
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
      output := Some m
    |}] ++ sufX'')
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m)
  [{|
     l := existT x li;
     input := input;
     destination :=
       equivocators_state_project IM eqv_descriptors''
         (state_update (equivocator_IM IM)
            (finite_trace_last is pre) x si');
     output := Some m
   |}] clear Hpr_pre_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX'': list (composite_transition_item IM)
eqv_descriptors'': equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si');
      output := Some m
    |}] ++ sufX'')
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre: equivocators_trace_project IM
  (equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0)) pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
      output := Some m
    |}] ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++
      [{|
         l := existT x li;
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
       |}] ++ sufX''))
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
      output := Some m
    |}] ++ sufX'')
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
Exists (field_selector output m)
  [{|
     l := existT x li;
     input := input;
     destination :=
       equivocators_state_project IM eqv_descriptors''
         (state_update (equivocator_IM IM)
            (finite_trace_last is pre) x si');
     output := Some m
   |}]
  constructor.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
x: index
input: option message
pre, suf: list transition_item
m: message
si': state
li: vlabel (IM x)
Htr: finite_valid_trace XE is
  (pre ++
   {|
     l := existT x (ContinueWith 0 li);
     input := input;
     destination :=
       state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
     output := Some m
   |} :: suf)
Hitem_not_equiv: x ∉ equivocating
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf))
initial_descriptors: equivocator_descriptors IM
preX', sufX'': list (composite_transition_item IM)
eqv_descriptors'': equivocator_descriptors IM
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l :=
         existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si';
        output := Some m
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update
         (equivocator_IM IM)
         (finite_trace_last is pre) x
         si');
      output := Some m
    |}] ++ sufX'')
Hpr_suf': equivocators_trace_project IM descriptors
  suf = Some (sufX'', eqv_descriptors'')
Hpr_pre: equivocators_trace_project IM
  (equivocator_descriptors_update IM
     eqv_descriptors'' x 
     (Existing 0)) pre =
Some (preX', initial_descriptors)
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
      output := Some m
    |}] ++ sufX'')
Hpre'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) is pre
Hsuf'_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     [{|
        l := existT x (ContinueWith 0 li);
        input := input;
        destination :=
         state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si';
        output := Some m
      |}]) suf
Hti: vtransition 
  (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input) =
(si', Some m)
Hv: vvalid (equivocator_IM IM x) 
  (ContinueWith 0 li)
  (finite_trace_last is pre x, input)
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is)
     (preX' ++
      [{|
         l := existT x li;
         input := input;
         destination :=
         equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
         output := Some m
       |}] ++ sufX''))
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is)
  (preX' ++
   [{|
      l := existT x li;
      input := input;
      destination :=
        equivocators_state_project IM
         eqv_descriptors''
         (state_update 
         (equivocator_IM IM)
         (finite_trace_last is pre) x si');
      output := Some m
    |}] ++ sufX'')
Hsingleton_d_item: is_singleton_state (IM x) si'
Heqv_descriptors'': eqv_descriptors'' x = Existing 0
field_selector output m
  {|
    l := existT x li;
    input := input;
    destination :=
      equivocators_state_project IM eqv_descriptors''
        (state_update (equivocator_IM IM)
           (finite_trace_last is pre) x si');
    output := Some m
  |} reflexivity.
  Unshelve.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
HasBeenReceivedCapability Free
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
HasBeenSentCapability Free
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
HasBeenReceivedCapability Free assumption.
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
item: transition_item
Hitem: item ∈ tr
Hitem_not_equiv: projT1 (l item) ∉ equivocating
m: message
Hm: field_selector output m item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX_Free: finite_valid_trace
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
HtrX_Pre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_preX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
HasBeenSentCapability Free assumption.
Qed.

Consider a valid_trace for the composition of equivocators with no message equivocation and fixed state equivocation.

Because any of its projections to the composition of original nodes contains all transitions from nodes not allowed to equivocate, then a final state of such a projection will be able to observe all messages sent or received by non-equivocating nodes in the initial trace.

Therefore if seeding the composition of equivocating nodes with these messages, the restriction of the initial trace to only the equivocating nodes will satisfy the trace_sub_item_input_is_seeded_or_sub_previously_sent property w.r.t. these messages, a sufficient condition for it being valid (finite_valid_trace_sub_projection).

Lemma equivocators_trace_sub_item_input_is_seeded_or_sub_previously_sent
  (is : vstate XE)
  (tr : list (vtransition_item XE))
  (s := finite_trace_last is tr)
  (Htr : finite_valid_trace XE is tr)
  (descriptors: equivocator_descriptors IM)
  (Hproper: proper_fixed_equivocator_descriptors descriptors s)
  (lst_trX := equivocators_state_project IM descriptors s)
  : trace_sub_item_input_is_seeded_or_sub_previously_sent
    (equivocator_IM IM) equivocating
    (no_additional_equivocations (free_composite_vlsm IM) lst_trX) tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) equivocating
  (no_additional_equivocations
     (free_composite_vlsm IM) lst_trX) tr
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) equivocating
  (no_additional_equivocations
     (free_composite_vlsm IM) lst_trX) tr
  intros pre item suf m Heq Hm Hitem.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item)
  destruct (free_equivocators_valid_trace_project descriptors is tr Hproper Htr)
    as [trX [initial_descriptors [Hinitial_descriptors [Htr_project [Hfinal_state HtrXFree]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item)
  assert (HtrXPre : finite_valid_trace (pre_loaded_with_all_messages_vlsm (free_composite_vlsm IM)) (equivocators_state_project IM initial_descriptors is) trX).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM initial_descriptors
     is) trX
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
no_additional_equivocations 
  (free_composite_vlsm IM) lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection 
         (equivocator_IM IM) equivocating pre_item)
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM initial_descriptors
     is) trX revert HtrXFree.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM initial_descriptors
     is) trX
→ finite_valid_trace
    (pre_loaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (equivocators_state_project IM initial_descriptors
       is) trX apply VLSM_incl_finite_valid_trace.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
VLSM_incl_part
  (composite_vlsm_machine IM (free_constraint IM))
  (pre_loaded_with_all_messages_vlsm_machine
     (free_composite_vlsm IM))
    apply vlsm_incl_pre_loaded_with_all_messages_vlsm.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item)
  assert (Hlst_trX : valid_state_prop (pre_loaded_with_all_messages_vlsm Free) lst_trX).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free) lst_trX
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
no_additional_equivocations 
  (free_composite_vlsm IM) lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection 
         (equivocator_IM IM) equivocating pre_item)
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free) lst_trX subst lst_trX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM descriptors s) subst s.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM descriptors
     (finite_trace_last is tr)) simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM descriptors
     (finite_trace_last is tr)) simpl in Hfinal_state.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM descriptors
     (finite_trace_last is tr)) rewrite Hfinal_state.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
    apply finite_valid_trace_last_pstate.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM initial_descriptors
     is) trX apply HtrXPre.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item)
  assert (Htr_free : finite_valid_trace  (pre_loaded_with_all_messages_vlsm FreeE) is tr).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE) is tr
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
no_additional_equivocations 
  (free_composite_vlsm IM) lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection 
         (equivocator_IM IM) equivocating pre_item)
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE) is tr revert Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
finite_valid_trace XE is tr
→ finite_valid_trace
    (pre_loaded_with_all_messages_vlsm FreeE) is tr  apply VLSM_incl_finite_valid_trace.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
VLSM_incl_part
  (composite_vlsm_machine (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint IM
        Hbs index_listing equivocating))
  (pre_loaded_with_all_messages_vlsm_machine FreeE)
    apply VLSM_incl_trans with (machine FreeE)
    ; [apply equivocators_fixed_equivocations_vlsm_incl_free|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
VLSM_incl (mk_vlsm (machine FreeE))
  (mk_vlsm
     {|
       transition := vtransition FreeE;
       valid := vvalid FreeE
     |})
    apply vlsm_incl_pre_loaded_with_all_messages_vlsm.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
tr: list (vtransition_item XE)
s:= finite_trace_last is tr: state
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Heq: tr = pre ++ [item] ++ suf
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is tr
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item)
  subst tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item)
  destruct Htr as [Htr His].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  (pre ++ [item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item)
  apply finite_valid_trace_from_app_iff in Htr as Hsuf.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  (pre ++ [item] ++ suf)
Hsuf: finite_valid_trace_from XE is pre
∧ finite_valid_trace_from XE
    (finite_trace_last is pre) 
    ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item) destruct Hsuf as [_ Hsuf].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  (pre ++ [item] ++ suf)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item)
  rewrite app_assoc in Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item)
  apply finite_valid_trace_from_app_iff in Htr as Hpre.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is (pre ++ [item])
∧ finite_valid_trace_from XE
    (finite_trace_last is (pre ++ [item])) suf
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item) destruct Hpre as [Hpre _].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is (pre ++ [item])
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item)
  apply finite_valid_trace_from_app_iff in Hpre.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
∧ finite_valid_trace_from XE
    (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item) destruct Hpre as [Hpre Hivt].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item)
  apply first_transition_valid in Hivt.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
no_additional_equivocations (free_composite_vlsm IM)
  lst_trX m
∨ (∃ pre_item : transition_item,
     pre_item ∈ pre
     ∧ output pre_item = Some m
       ∧ from_sub_projection (equivocator_IM IM)
           equivocating pre_item)
  destruct (Free_no_additional_equivocation_decidable lst_trX m)
  ; [left; assumption|right].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ no_additional_equivocations Free lst_trX m
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  unfold no_additional_equivocations in n.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  assert (Hsuf_free : finite_valid_trace_from (pre_loaded_with_all_messages_vlsm FreeE) (finite_trace_last is pre) ([item] ++ suf)).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre) ([item] ++ suf)
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([item] ++ suf)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection 
        (equivocator_IM IM) equivocating pre_item
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre) ([item] ++ suf) revert Hsuf.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
finite_valid_trace_from XE (finite_trace_last is pre)
  ([item] ++ suf)
→ finite_valid_trace_from
    (pre_loaded_with_all_messages_vlsm FreeE)
    (finite_trace_last is pre) ([item] ++ suf) apply VLSM_incl_finite_valid_trace_from.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
VLSM_incl_part
  (composite_vlsm_machine (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint IM
        Hbs index_listing equivocating))
  (pre_loaded_with_all_messages_vlsm_machine FreeE)
    apply VLSM_incl_trans with (machine FreeE)
    ; [apply equivocators_fixed_equivocations_vlsm_incl_free|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
VLSM_incl (mk_vlsm (machine FreeE))
  (mk_vlsm
     {|
       transition := vtransition FreeE;
       valid := vvalid FreeE
     |})
    apply vlsm_incl_pre_loaded_with_all_messages_vlsm.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([item] ++ suf)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  assert (Hs_free : valid_state_prop  (pre_loaded_with_all_messages_vlsm FreeE) s).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([item] ++ suf)
valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE) s
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([item] ++ suf)
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection 
        (equivocator_IM IM) equivocating pre_item
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([item] ++ suf)
valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE) s apply finite_valid_trace_last_pstate in Hsuf_free.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
Hsuf_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     ([item] ++ suf))
valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE) s subst s.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)): vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
Hsuf_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     ([item] ++ suf))
valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is (pre ++ [item] ++ suf))
    rewrite finite_trace_last_app.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)): vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
Hsuf_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last
     (finite_trace_last is pre)
     ([item] ++ suf))
valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last (finite_trace_last is pre)
     ([item] ++ suf)) assumption.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE 
  (l item)
  (finite_trace_last is pre, input item)
  (destination item, output item)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre) 
  ([item] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is (pre ++ [item] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input item = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating item
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors 
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre ++ [item] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([item] ++ suf)
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  destruct item as (l, iom, s0, oom).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
iom: option message
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |})
  (destination
     {|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |},
  output
    {|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |})
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: input
  {|
    l := l;
    input := iom;
    destination := s0;
    output := oom
  |} = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := iom;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := iom;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf)
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  simpl in Hm.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
iom: option message
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hivt: input_valid_transition XE
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |})
  (destination
     {|
       l := l;
       input := iom;
       destination := s0;
       output := oom
     |},
  output
    {|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |})
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: iom = Some m
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := iom;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := iom;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf)
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := iom;
      destination := s0;
      output := oom
    |}] ++ suf)
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item subst iom.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hivt: input_valid_transition XE
  (VLSM.l
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |})
  (destination
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |},
  output
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |})
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  destruct Hivt as [[_ [_ [_ [[Hc _] Hfixed]]]] Ht].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_no_equivocations 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (VLSM.l
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |})
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating
  (composite_transition 
     (equivocator_IM IM)
     (VLSM.l
        {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
        |})
     (finite_trace_last is pre,
     input
       {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |})).1
Ht: transition
  (VLSM.l
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}) =
(destination
   {|
     l := l;
     input := Some m;
     destination := s0;
     output := oom
   |},
output
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |})
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  simpl in Ht, Hfixed.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_no_equivocations 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (VLSM.l
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |})
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating
  (let (i, li) := l in
   let (si', om') :=
     vtransition 
       (equivocator_IM IM i) li
       (finite_trace_last is pre i, Some m) in
   (state_update 
      (equivocator_IM IM)
      (finite_trace_last is pre) i si', om')).1
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is pre i, Some m) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is pre) i si', om')) =
(s0, oom)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item rewrite Ht in Hfixed.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_no_equivocations 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (VLSM.l
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |})
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating 
  (s0, oom).1
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is pre i, Some m) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is pre) i si', om')) =
(s0, oom)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item simpl in Hfixed.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_no_equivocations 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (VLSM.l
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |})
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
Ht: (let (i, li) := l in
 let (si', om') :=
   vtransition 
     (equivocator_IM IM i) li
     (finite_trace_last is pre i, Some m) in
 (state_update 
    (equivocator_IM IM)
    (finite_trace_last is pre) i si', om')) =
(s0, oom)
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  clear Ht.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_no_equivocations 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (VLSM.l
     {|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |})
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |})
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  destruct Hc as [Hc | Hinit]; [|contradiction].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  assert (Hpre_free : finite_valid_trace FreeE is pre).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
finite_valid_trace FreeE is pre
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection 
        (equivocator_IM IM) equivocating pre_item
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
finite_valid_trace FreeE is pre split; [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
finite_valid_trace_from FreeE is pre revert Hpre.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
finite_valid_trace_from XE is pre
→ finite_valid_trace_from FreeE is pre apply VLSM_incl_finite_valid_trace_from.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
VLSM_incl_part
  (composite_vlsm_machine (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint IM
        Hbs index_listing equivocating))
  (composite_vlsm_machine (equivocator_IM IM)
     (free_constraint (equivocator_IM IM)))
    apply equivocators_fixed_equivocations_vlsm_incl_free.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  assert (Hpre_lst_free : valid_state_prop FreeE (finite_trace_last is pre)).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
valid_state_prop FreeE (finite_trace_last is pre)
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection 
        (equivocator_IM IM) equivocating pre_item
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
valid_state_prop FreeE (finite_trace_last is pre) apply (finite_valid_trace_last_pstate FreeE).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
finite_valid_trace_from FreeE is pre apply Hpre_free. }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  specialize (specialized_proper_sent_rev FreeE _ Hpre_lst_free m) as Hproper_sent.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: composite_has_been_sent 
  (equivocator_IM IM) 
  (equivocator_Hbs IM Hbs)
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}).1 m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hproper_sent: has_been_sent FreeE
  (finite_trace_last is pre) m
→ specialized_selected_message_exists_in_all_traces
    FreeE 
    (field_selector output)
    (finite_trace_last is pre) m
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  apply Hproper_sent in Hc.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: specialized_selected_message_exists_in_all_traces
  FreeE (field_selector output)
  (finite_trace_last is pre) m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hproper_sent: has_been_sent FreeE
  (finite_trace_last is pre) m
→ specialized_selected_message_exists_in_all_traces
    FreeE 
    (field_selector output)
    (finite_trace_last is pre) m
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item clear Hproper_sent.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: specialized_selected_message_exists_in_all_traces
  FreeE (field_selector output)
  (finite_trace_last is pre) m
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  specialize (Hc  is pre (valid_trace_add_default_last Hpre_free)).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: trace_has_message (field_selector output) m pre
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  apply Exists_exists in Hc.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hc: ∃ x : transition_item,
  x ∈ pre ∧ field_selector output m x
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  destruct Hc as [pre_item [Hpre_item Hpre_m]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∃ pre_item : transition_item,
  pre_item ∈ pre
  ∧ output pre_item = Some m
    ∧ from_sub_projection (equivocator_IM IM)
        equivocating pre_item
  exists pre_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
pre_item ∈ pre
∧ output pre_item = Some m
  ∧ from_sub_projection (equivocator_IM IM)
      equivocating pre_item split; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
output pre_item = Some m
∧ from_sub_projection (equivocator_IM IM) equivocating
    pre_item split; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: equivocators_trace_project IM
  descriptors
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (trX, initial_descriptors)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
from_sub_projection (equivocator_IM IM) equivocating
  pre_item

  apply equivocators_trace_project_app_iff in Htr_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
Htr_project: ∃ (preX
   sufX : 
   list (composite_transition_item IM)) 
  (eqv_descriptors' : 
   equivocator_descriptors IM),
  equivocators_trace_project IM
    descriptors
    ([{|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |}] ++ suf) =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project IM
      eqv_descriptors' pre =
    Some (preX, initial_descriptors)
    ∧ trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
from_sub_projection (equivocator_IM IM) equivocating
  pre_item
  destruct Htr_project as [preX [sufX [final_descriptors' [Hsuf_project [Htr_project Heq]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
from_sub_projection (equivocator_IM IM) equivocating
  pre_item

  assert (Hfinal' : proper_fixed_equivocator_descriptors final_descriptors'  (finite_trace_last is pre)).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_fixed_equivocator_descriptors
  final_descriptors' (finite_trace_last is pre)
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
from_sub_projection 
  (equivocator_IM IM) equivocating pre_item
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_fixed_equivocator_descriptors
  final_descriptors' (finite_trace_last is pre) split.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre)
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∀ i : index,
  i ∉ equivocating → final_descriptors' i = Existing 0
    -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre) apply proj1 in Hproper.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre) subst s.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)): vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre) rewrite finite_trace_last_app in Hproper.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  descriptors
  (finite_trace_last
     (finite_trace_last is pre)
     ([{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)): vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre)
      destruct (preloaded_equivocators_valid_trace_from_project IM _ _ _ Hproper Hsuf_free)
        as [_sufX [_final_descriptors' [_Hsuf_project [Hproper' _]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  descriptors
  (finite_trace_last
     (finite_trace_last is pre)
     ([{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)): vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
_sufX: list (composite_transition_item IM)
_final_descriptors': equivocator_descriptors IM
_Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (_sufX, _final_descriptors')
Hproper': proper_equivocator_descriptors IM
  _final_descriptors'
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre)
      rewrite Hsuf_project in _Hsuf_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  descriptors
  (finite_trace_last
     (finite_trace_last is pre)
     ([{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)): vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
_sufX: list (composite_transition_item IM)
_final_descriptors': equivocator_descriptors IM
_Hsuf_project: Some (sufX, final_descriptors') =
Some (_sufX, _final_descriptors')
Hproper': proper_equivocator_descriptors IM
  _final_descriptors'
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre)
      inversion _Hsuf_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  descriptors
  (finite_trace_last
     (finite_trace_last is pre)
     ([{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)): vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
_sufX: list (composite_transition_item IM)
_final_descriptors': equivocator_descriptors IM
_Hsuf_project: Some (sufX, final_descriptors') =
Some (_sufX, _final_descriptors')
Hproper': proper_equivocator_descriptors IM
  _final_descriptors'
  (finite_trace_last is pre)
H0: sufX = _sufX
H1: final_descriptors' = _final_descriptors'
proper_equivocator_descriptors IM _final_descriptors'
  (finite_trace_last is pre) subst _sufX _final_descriptors'.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  descriptors
  (finite_trace_last
     (finite_trace_last is pre)
     ([{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)): vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is pre)
_Hsuf_project: Some (sufX, final_descriptors') =
Some (sufX, final_descriptors')
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre) clear _Hsuf_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_equivocator_descriptors IM
  descriptors
  (finite_trace_last
     (finite_trace_last is pre)
     ([{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
lst_trX:= equivocators_state_project IM descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)): vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hproper': proper_equivocator_descriptors IM
  final_descriptors'
  (finite_trace_last is pre)
proper_equivocator_descriptors IM final_descriptors'
  (finite_trace_last is pre)
      assumption.
    -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
∀ i : index,
  i ∉ equivocating → final_descriptors' i = Existing 0 specialize (equivocators_trace_project_preserves_zero_descriptors IM _ _ Hsuf_free descriptors) as Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hpr: ∀ (idescriptors : equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  equivocators_trace_project IM descriptors
    ([{|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |}] ++ suf) = 
  Some (trX, idescriptors)
  → ∀ i : index,
      descriptors i = Existing 0
      → idescriptors i = Existing 0
∀ i : index,
  i ∉ equivocating → final_descriptors' i = Existing 0
      specialize (Hpr _ _ Hsuf_project).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hpr: ∀ i : index,
  descriptors i = Existing 0
  → final_descriptors' i = Existing 0
∀ i : index,
  i ∉ equivocating → final_descriptors' i = Existing 0
      intros i Hi.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hpr: ∀ i : index,
  descriptors i = Existing 0
  → final_descriptors' i = Existing 0
i: index
Hi: i ∉ equivocating
final_descriptors' i = Existing 0
      apply Hpr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hpr: ∀ i : index,
  descriptors i = Existing 0
  → final_descriptors' i = Existing 0
i: index
Hi: i ∉ equivocating
descriptors i = Existing 0 apply (proj2 Hproper).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hpr: ∀ i : index,
  descriptors i = Existing 0
  → final_descriptors' i = Existing 0
i: index
Hi: i ∉ equivocating
i ∉ equivocating assumption.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
from_sub_projection (equivocator_IM IM) equivocating
  pre_item
  assert
    (Hfutures : in_futures (pre_loaded_with_all_messages_vlsm FreeE)
      (destination pre_item) s0).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: in_futures
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
from_sub_projection 
  (equivocator_IM IM) equivocating pre_item
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0 apply elem_of_list_split in Hpre_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: ∃ l1 l2 : list transition_item,
  pre = l1 ++ pre_item :: l2
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    destruct Hpre_item as [pre_pre [pre_suf Hpre_item]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = pre_pre ++ pre_item :: pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    change (pre_item :: pre_suf) with ([pre_item] ++ pre_suf) in Hpre_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = pre_pre ++ [pre_item] ++ pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    rewrite app_assoc in Hpre_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = (pre_pre ++ [pre_item]) ++ pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    rewrite app_assoc in Htr_free.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = (pre_pre ++ [pre_item]) ++ pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    destruct Htr_free as [Htr_free _].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = (pre_pre ++ [pre_item]) ++ pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    apply finite_valid_trace_from_app_iff in Htr_free.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = (pre_pre ++ [pre_item]) ++ pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
∧ finite_valid_trace_from
    (pre_loaded_with_all_messages_vlsm
       FreeE)
    (finite_trace_last is
       (pre ++
        [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}])) suf
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    destruct Htr_free as [Htr_s0 _].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hpre_item: pre = (pre_pre ++ [pre_item]) ++ pre_suf
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_s0: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    subst pre.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is
     ((pre_pre ++ [pre_item]) ++ pre_suf))
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre: finite_valid_trace_from XE is
  ((pre_pre ++ [pre_item]) ++ pre_suf)
Htr: finite_valid_trace_from XE is
  ((((pre_pre ++ [pre_item]) ++ pre_suf) ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (((pre_pre ++ [pre_item]) ++ pre_suf) ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors'
  ((pre_pre ++ [pre_item]) ++ pre_suf) =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (((pre_pre ++ [pre_item]) ++
       pre_suf) ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is
     ((pre_pre ++ [pre_item]) ++ pre_suf))
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is
     ((pre_pre ++ [pre_item]) ++
      pre_suf))
Hpre_free: finite_valid_trace FreeE is
  ((pre_pre ++ [pre_item]) ++ pre_suf)
Htr_s0: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (((pre_pre ++ [pre_item]) ++ pre_suf) ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
n: ¬ has_been_observed Free lst_trX m
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     ((pre_pre ++ [pre_item]) ++ pre_suf))
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0 clear -Htr_s0.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: set index
index_listing: list index
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: vstate XE
l: label
s0: state
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Htr_s0: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (((pre_pre ++ [pre_item]) ++ pre_suf) ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    rewrite <- app_assoc in Htr_s0.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: set index
index_listing: list index
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: vstate XE
l: label
s0: state
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Htr_s0: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  ((pre_pre ++ [pre_item]) ++
   pre_suf ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    apply (finite_valid_trace_from_app_iff (pre_loaded_with_all_messages_vlsm FreeE)) in Htr_s0.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: set index
index_listing: list index
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: vstate XE
l: label
s0: state
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Htr_s0: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  is (pre_pre ++ [pre_item])
∧ finite_valid_trace_from
    (pre_loaded_with_all_messages_vlsm FreeE)
    (finite_trace_last is
       (pre_pre ++ [pre_item]))
    (pre_suf ++
     [{|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |}])
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    destruct Htr_s0 as [_ Hfuture].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: set index
index_listing: list index
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: vstate XE
l: label
s0: state
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hfuture: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre_pre ++ [pre_item]))
  (pre_suf ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    rewrite finite_trace_last_is_last in Hfuture.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: set index
index_listing: list index
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: vstate XE
l: label
s0: state
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hfuture: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item)
  (pre_suf ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
in_futures (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
    eexists.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: set index
index_listing: list index
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: vstate XE
l: label
s0: state
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hfuture: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item)
  (pre_suf ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
finite_valid_trace_from_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0 ?tr
    apply valid_trace_add_last;[apply Hfuture|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: set index
index_listing: list index
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
is: vstate XE
l: label
s0: state
oom: option message
m: message
pre_item: transition_item
pre_pre, pre_suf: list transition_item
Hfuture: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item)
  (pre_suf ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}])
finite_trace_last (destination pre_item)
  (pre_suf ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}]) = s0
    rewrite finite_trace_last_is_last;reflexivity.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: in_futures
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination pre_item) s0
from_sub_projection (equivocator_IM IM) equivocating
  pre_item
  apply (@in_futures_reflects_fixed_equivocation _ _ _ IM index_listing equivocating)
    in Hfutures
  ; [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
from_sub_projection (equivocator_IM IM) equivocating
  pre_item
  destruct (free_equivocators_valid_trace_project final_descriptors' is pre Hfinal' (conj Hpre His))
    as [_preX [_initial_descriptors [_ [_Htr_project [Hpre_final_state _]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
_preX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (_preX, _initial_descriptors)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     _initial_descriptors is) _preX
from_sub_projection (equivocator_IM IM) equivocating
  pre_item
  rewrite Htr_project in _Htr_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
_preX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Htr_project: Some (preX, initial_descriptors) =
Some (_preX, _initial_descriptors)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     _initial_descriptors is) _preX
from_sub_projection (equivocator_IM IM) equivocating
  pre_item inversion _Htr_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
_preX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Htr_project: Some (preX, initial_descriptors) =
Some (_preX, _initial_descriptors)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     _initial_descriptors is) _preX
H0: preX = _preX
H1: initial_descriptors = _initial_descriptors
from_sub_projection (equivocator_IM IM) equivocating
  pre_item subst _preX _initial_descriptors.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
_Htr_project: Some (preX, initial_descriptors) =
Some (preX, initial_descriptors)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
from_sub_projection (equivocator_IM IM) equivocating
  pre_item
  clear _Htr_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
from_sub_projection (equivocator_IM IM) equivocating
  pre_item
  unfold from_sub_projection, pre_VLSM_projection_in_projection,
    composite_label_sub_projection_option.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
is_Some
  match
    decide (projT1 (VLSM.l pre_item) ∈ equivocating)
  with
  | left i_in =>
      Some
        (composite_label_sub_projection
           (equivocator_IM IM) equivocating
           (VLSM.l pre_item) i_in)
  | right _ => None
  end
  case_decide; [eexists; reflexivity|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
is_Some None
  elim n.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
has_been_observed Free lst_trX m
  specialize
    (not_equivocating_sent_message_has_been_observed_in_projection
      is pre (conj Hpre His) pre_item Hpre_item H m Hpre_m final_descriptors'
      Hfinal'
    )
    as Hobs_m.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
Hobs_m: has_been_observed Free
  (equivocators_state_project IM
     final_descriptors'
     (finite_trace_last is pre)) m
has_been_observed Free lst_trX m
  rewrite Hpre_final_state in Hobs_m.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
Hobs_m: has_been_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX) m
has_been_observed Free lst_trX m

  revert Hobs_m.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
has_been_observed Free
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX) m
→ has_been_observed Free lst_trX m
  apply in_futures_preserving_oracle_from_stepwise with item_sends_or_receives.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
oracle_stepwise_props item_sends_or_receives
  (has_been_observed Free)
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
in_futures 
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX) lst_trX
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
oracle_stepwise_props item_sends_or_receives
  (has_been_observed Free) apply has_been_observed_stepwise_props.
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
lst_trX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free lst_trX m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  lst_trX
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
in_futures (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX) lst_trX subst lst_trX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
s:= finite_trace_last is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf): state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  s
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free
    (equivocators_state_project IM descriptors s)
    m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     descriptors s)
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
in_futures (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX)
  (equivocators_state_project IM descriptors s) subst s.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free
    (equivocators_state_project IM descriptors
       (finite_trace_last is
         (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf))) m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf)))
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
in_futures (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX)
  (equivocators_state_project IM descriptors
     (finite_trace_last is
        (pre ++
         [{|
            l := l;
            input := Some m;
            destination := s0;
            output := oom
          |}] ++ suf))) simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free
    (equivocators_state_project IM descriptors
       (finite_trace_last is
         (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf))) m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf)))
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
in_futures (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX)
  (equivocators_state_project IM descriptors
     (finite_trace_last is
        (pre ++
         {|
           l := l;
           input := Some m;
           destination := s0;
           output := oom
         |} :: suf))) simpl in Hfinal_state.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free
    (equivocators_state_project IM descriptors
       (finite_trace_last is
         (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf))) m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf)))
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
in_futures (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX)
  (equivocators_state_project IM descriptors
     (finite_trace_last is
        (pre ++
         {|
           l := l;
           input := Some m;
           destination := s0;
           output := oom
         |} :: suf)))
    rewrite Hfinal_state.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
Heq: trX = preX ++ sufX
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) trX
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free
    (equivocators_state_project IM descriptors
       (finite_trace_last is
         (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf))) m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf)))
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
in_futures (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) subst trX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX ++ sufX)
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free
    (equivocators_state_project IM descriptors
       (finite_trace_last is
         (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf))) m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf)))
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
in_futures (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) (preX ++ sufX))
    rewrite finite_trace_last_app.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX ++ sufX)
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free
    (equivocators_state_project IM descriptors
       (finite_trace_last is
         (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf))) m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf)))
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
in_futures (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX)
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) preX) sufX)
    exists sufX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: vstate XE
pre: list
  (composite_transition_item (equivocator_IM IM))
l: label
s0: state
oom: option message
suf: list
  (composite_transition_item (equivocator_IM IM))
m: message
Htr: finite_valid_trace_from XE is
  ((pre ++
    [{|
       l := l;
       input := Some m;
       destination := s0;
       output := oom
     |}]) ++ suf)
Hpre: finite_valid_trace_from XE is pre
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
pre_item: transition_item
Hpre_item: pre_item ∈ pre
Hpre_m: field_selector output m pre_item
Hfixed: state_has_fixed_equivocation IM index_listing
  equivocating s0
His: initial_state_prop is
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hitem: from_sub_projection 
  (equivocator_IM IM) equivocating
  {|
    l := l;
    input := Some m;
    destination := s0;
    output := oom
  |}
initial_descriptors: equivocator_descriptors IM
Hinitial_descriptors: proper_fixed_equivocator_descriptors
  initial_descriptors is
preX, sufX: list (composite_transition_item IM)
final_descriptors': equivocator_descriptors IM
Hsuf_project: equivocators_trace_project IM
  descriptors
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf) =
Some (sufX, final_descriptors')
Htr_project: equivocators_trace_project IM
  final_descriptors' pre =
Some (preX, initial_descriptors)
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
HtrXFree: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
Hfinal_state: equivocators_state_project IM
  descriptors
  (finite_trace_last is
     (pre ++
      {|
        l := l;
        input := Some m;
        destination := s0;
        output := oom
      |} :: suf)) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is)
  (preX ++ sufX)
Hs_free: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is
     (pre ++
      [{|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
       |}] ++ suf))
Hsuf_free: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (finite_trace_last is pre)
  ([{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
n: ¬ has_been_observed Free
    (equivocators_state_project IM descriptors
       (finite_trace_last is
         (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf))) m
Hlst_trX: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     descriptors
     (finite_trace_last is
        (pre ++
         [
         {|
         l := l;
         input := Some m;
         destination := s0;
         output := oom
         |}] ++ suf)))
Htr_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  is
  (pre ++
   [{|
      l := l;
      input := Some m;
      destination := s0;
      output := oom
    |}] ++ suf)
Hpre_free: finite_valid_trace FreeE is pre
Hpre_lst_free: valid_state_prop FreeE
  (finite_trace_last is pre)
Hfinal': proper_fixed_equivocator_descriptors
  final_descriptors'
  (finite_trace_last is pre)
Hfutures: state_has_fixed_equivocation IM
  index_listing equivocating
  (destination pre_item)
Hpre_final_state: equivocators_state_project IM
  final_descriptors'
  (finite_trace_last is pre) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) preX
H: projT1 (VLSM.l pre_item) ∉ equivocating
finite_valid_trace_from_to
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX)
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) preX) sufX) sufX
    clear -HtrXPre.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: set index
index_listing: list index
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
Free:= free_composite_vlsm IM: VLSM message
is: vstate XE
initial_descriptors: equivocator_descriptors IM
preX, sufX: list (composite_transition_item IM)
HtrXPre: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
finite_valid_trace_from_to
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX)
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) preX) sufX) sufX
    apply proj1 in HtrXPre.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: set index
index_listing: list index
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
Free:= free_composite_vlsm IM: VLSM message
is: vstate XE
initial_descriptors: equivocator_descriptors IM
preX, sufX: list (composite_transition_item IM)
HtrXPre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) 
  (preX ++ sufX)
finite_valid_trace_from_to
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX)
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) preX) sufX) sufX
    apply finite_valid_trace_from_app_iff in HtrXPre.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: set index
index_listing: list index
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
Free:= free_composite_vlsm IM: VLSM message
is: vstate XE
initial_descriptors: equivocator_descriptors IM
preX, sufX: list (composite_transition_item IM)
HtrXPre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) preX
∧ finite_valid_trace_from
    (pre_loaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) preX) sufX
finite_valid_trace_from_to
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX)
  (finite_trace_last
     (finite_trace_last
        (equivocators_state_project IM
           initial_descriptors is) preX) sufX) sufX
    apply valid_trace_add_default_last.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
equivocating: set index
index_listing: list index
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
Free:= free_composite_vlsm IM: VLSM message
is: vstate XE
initial_descriptors: equivocator_descriptors IM
preX, sufX: list (composite_transition_item IM)
HtrXPre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm IM))
  (equivocators_state_project IM
     initial_descriptors is) preX
∧ finite_valid_trace_from
    (pre_loaded_with_all_messages_vlsm
       (free_composite_vlsm IM))
    (finite_trace_last
       (equivocators_state_project IM
          initial_descriptors is) preX) sufX
finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) preX) sufX
    apply HtrXPre.
Qed.

Because any of its projections to the composition of original nodes contains all transitions from nodes not allowed to equivocate, all messages sent by non-equivocating nodes in the original trace will also be sent in any projection.

Lemma equivocator_vlsm_trace_project_reflect_non_equivocating
  (is: composite_state (equivocator_IM IM))
  (tr: list (composite_transition_item (equivocator_IM IM)))
  (Htr: finite_valid_trace XE is tr)
  (m: message)
  (final_descriptors: equivocator_descriptors IM)
  (Hproper: proper_fixed_equivocator_descriptors final_descriptors (finite_trace_last is tr))
  (trX: list (composite_transition_item IM))
  (initial_descriptors: equivocator_descriptors IM)
  (Htr_project: equivocators_trace_project IM final_descriptors tr = Some (trX, initial_descriptors))
  (item: composite_transition_item (equivocator_IM IM))
  (Hitem: item ∈ tr)
  (Houtput: output item = Some m)
  (Hno_equiv_item: (projT1 (l item)) ∉ equivocating)
  : trace_has_message (field_selector output) m trX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  apply elem_of_list_split in Hitem.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
item: composite_transition_item (equivocator_IM IM)
Hitem: ∃ l1
   l2 : list
         (composite_transition_item
         (equivocator_IM IM)),
  tr = l1 ++ item :: l2
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  destruct Hitem as [pre [suf Heq_tr]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heq_tr: tr = pre ++ item :: suf
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  change (item :: suf) with ([item] ++ suf) in Heq_tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is tr)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heq_tr: tr = pre ++ [item] ++ suf
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  subst tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors
  (pre ++ [item] ++ suf) =
Some (trX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  apply equivocators_trace_project_app_iff in Htr_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: ∃ (preX
   sufX : 
   list (composite_transition_item IM)) 
  (eqv_descriptors' : 
   equivocator_descriptors IM),
  equivocators_trace_project IM
    final_descriptors 
    ([item] ++ suf) =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project IM
      eqv_descriptors' pre =
    Some (preX, initial_descriptors)
    ∧ trX = preX ++ sufX
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  destruct Htr_project as [preX [item_sufX [eqv_pre [Hpr_item_suf [Hpr_pre Heq_trX]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
preX, item_sufX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
Hpr_item_suf: equivocators_trace_project IM
  final_descriptors 
  ([item] ++ suf) =
Some (item_sufX, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Heq_trX: trX = preX ++ item_sufX
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  subst trX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX, item_sufX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
Hpr_item_suf: equivocators_trace_project IM
  final_descriptors 
  ([item] ++ suf) =
Some (item_sufX, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m
  (preX ++ item_sufX)
  apply Exists_app.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX, item_sufX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
Hpr_item_suf: equivocators_trace_project IM
  final_descriptors 
  ([item] ++ suf) =
Some (item_sufX, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) preX
∨ Exists (field_selector output m) item_sufX right.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX, item_sufX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
Hpr_item_suf: equivocators_trace_project IM
  final_descriptors 
  ([item] ++ suf) =
Some (item_sufX, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) item_sufX
  apply equivocators_trace_project_app_iff in Hpr_item_suf.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX, item_sufX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
Hpr_item_suf: ∃ (preX
   sufX : 
   list (composite_transition_item IM)) 
  (eqv_descriptors' : 
   equivocator_descriptors IM),
  equivocators_trace_project IM
    final_descriptors suf =
  Some (sufX, eqv_descriptors')
  ∧ equivocators_trace_project IM
      eqv_descriptors' [item] =
    Some (preX, eqv_pre)
    ∧ item_sufX = preX ++ sufX
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) item_sufX
  destruct Hpr_item_suf as [itemXs [sufX [eqv_item [Hpr_suf [Hpr_item Heq_item_sufX]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX, item_sufX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Heq_item_sufX: item_sufX = itemXs ++ sufX
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) item_sufX
  subst item_sufX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) (itemXs ++ sufX)
  apply Exists_app.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
∨ Exists (field_selector output m) sufX left.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  destruct Htr as [Htr _].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  (pre ++ [item] ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  rewrite app_assoc in Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is
  ((pre ++ [item]) ++ suf)
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  apply finite_valid_trace_from_app_iff in Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_from XE is (pre ++ [item])
∧ finite_valid_trace_from XE
    (finite_trace_last is (pre ++ [item])) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  destruct Htr as [Hpre Hsuf].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hpre: finite_valid_trace_from XE is (pre ++ [item])
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is (pre ++ [item])) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  apply finite_valid_trace_from_app_iff in Hpre.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hpre: finite_valid_trace_from XE is pre
∧ finite_valid_trace_from XE
    (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is (pre ++ [item])) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  destruct Hpre as [_ Hitem].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is (pre ++ [item])) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last is
     (pre ++ [item] ++ suf))
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  rewrite app_assoc,finite_trace_last_app in Hproper.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE
  (finite_trace_last is (pre ++ [item])) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (finite_trace_last is (pre ++ [item]))
     suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  rewrite finite_trace_last_is_last in Hsuf, Hproper.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Exists (field_selector output m) itemXs
  assert (Hsufpre : finite_valid_trace_from (pre_loaded_with_all_messages_vlsm FreeE) (destination item) suf).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
Exists (field_selector output m) itemXs
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
    revert Hsuf.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
finite_valid_trace_from XE (destination item) suf
→ finite_valid_trace_from
    (pre_loaded_with_all_messages_vlsm FreeE)
    (destination item) suf apply VLSM_incl_finite_valid_trace_from.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
VLSM_incl_part
  (composite_vlsm_machine (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint IM
        Hbs index_listing equivocating))
  (pre_loaded_with_all_messages_vlsm_machine FreeE)
    apply VLSM_incl_trans with (machine FreeE).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
VLSM_incl
  (mk_vlsm
     {|
       transition :=
         composite_transition (equivocator_IM IM);
       valid :=
         constrained_composite_valid
           (equivocator_IM IM)
           (equivocators_fixed_equivocations_constraint
              IM Hbs index_listing equivocating)
     |}) (mk_vlsm (machine FreeE))
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
VLSM_incl 
  (mk_vlsm (machine FreeE))
  (mk_vlsm
     {|
       transition := vtransition FreeE;
       valid := vvalid FreeE
     |})
    -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
VLSM_incl
  (mk_vlsm
     {|
       transition :=
         composite_transition (equivocator_IM IM);
       valid :=
         constrained_composite_valid
           (equivocator_IM IM)
           (equivocators_fixed_equivocations_constraint
              IM Hbs index_listing equivocating)
     |}) (mk_vlsm (machine FreeE)) apply equivocators_fixed_equivocations_vlsm_incl_free.
    -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
VLSM_incl (mk_vlsm (machine FreeE))
  (mk_vlsm
     {|
       transition := vtransition FreeE;
       valid := vvalid FreeE
     |}) apply vlsm_incl_pre_loaded_with_all_messages_vlsm.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
Exists (field_selector output m) itemXs
  specialize
    (equivocators_trace_project_preserves_proper_fixed_equivocator_descriptors _ _ Hsufpre
      final_descriptors eqv_item _ Hpr_suf Hproper
    ) as Hproper_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
Hproper_item: proper_fixed_equivocator_descriptors
  eqv_item 
  (destination item)
Exists (field_selector output m) itemXs
  destruct Hproper_item as [Hproper_item Hproper_fixed_item].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item 
  (destination item)
Hproper_fixed_item: ∀ i : index,
  i ∉ equivocating
  → eqv_item i = Existing 0
Exists (field_selector output m) itemXs
  specialize (Hproper_fixed_item _ Hno_equiv_item).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item 
  (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Exists (field_selector output m) itemXs
  specialize
    (no_equivocating_equivocators_transition_item_project IM eqv_item item
      Hproper_fixed_item
    ) as Hex.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item 
  (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Hex: is_singleton_state 
  (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
→ ∀ s : composite_state (equivocator_IM IM),
    composite_valid 
      (equivocator_IM IM) 
      (l item) 
      (s, input item)
    → composite_transition 
        (equivocator_IM IM) 
        (l item) 
        (s, input item) =
      (destination item, output item)
      → ∃ Hex : 
         existing_equivocator_label
         (IM (projT1 (l item)))
         (projT2 (l item)),
         let lx :=
         existT 
         (projT1 (l item))
         (existing_equivocator_label_extract
         (IM (projT1 (l item)))
         (projT2 (l item)) Hex) in
         equivocators_transition_item_project IM
         eqv_item item =
         Some
         (
         Some
         {|
         l := lx;
         input := input item;
         destination :=
         equivocators_state_project IM eqv_item
         (destination item);
         output := output item
         |}, eqv_item)
Exists (field_selector output m) itemXs
  spec Hex.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item 
  (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Hex: is_singleton_state 
  (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
→ ∀ s : composite_state (equivocator_IM IM),
    composite_valid 
      (equivocator_IM IM) 
      (l item) 
      (s, input item)
    → composite_transition 
        (equivocator_IM IM) 
        (l item) 
        (s, input item) =
      (destination item, output item)
      → ∃ Hex : 
         existing_equivocator_label
         (IM (projT1 (l item)))
         (projT2 (l item)),
         let lx :=
         existT 
         (projT1 (l item))
         (existing_equivocator_label_extract
         (IM (projT1 (l item)))
         (projT2 (l item)) Hex) in
         equivocators_transition_item_project IM
         eqv_item item =
         Some
         (
         Some
         {|
         l := lx;
         input := input item;
         destination :=
         equivocators_state_project IM eqv_item
         (destination item);
         output := output item
         |}, eqv_item)
is_singleton_state (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item 
  (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Hex: ∀ s : composite_state (equivocator_IM IM),
  composite_valid 
    (equivocator_IM IM) 
    (l item) 
    (s, input item)
  → composite_transition 
      (equivocator_IM IM) 
      (l item) 
      (s, input item) =
    (destination item, output item)
    → ∃ Hex : 
         existing_equivocator_label
         (IM (projT1 (l item)))
         (projT2 (l item)),
        let lx :=
         existT 
         (projT1 (l item))
         (existing_equivocator_label_extract
         (IM (projT1 (l item)))
         (projT2 (l item)) Hex) in
        equivocators_transition_item_project IM
         eqv_item item =
        Some
         (
         Some
         {|
         l := lx;
         input := input item;
         destination :=
         equivocators_state_project IM eqv_item
         (destination item);
         output := output item
         |}, eqv_item)
Exists (field_selector output m) itemXs
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item 
  (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Hex: is_singleton_state 
  (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
→ ∀ s : composite_state (equivocator_IM IM),
    composite_valid 
      (equivocator_IM IM) 
      (l item) 
      (s, input item)
    → composite_transition 
        (equivocator_IM IM) 
        (l item) 
        (s, input item) =
      (destination item, output item)
      → ∃ Hex : 
         existing_equivocator_label
         (IM (projT1 (l item)))
         (projT2 (l item)),
         let lx :=
         existT 
         (projT1 (l item))
         (existing_equivocator_label_extract
         (IM (projT1 (l item)))
         (projT2 (l item)) Hex) in
         equivocators_transition_item_project IM
         eqv_item item =
         Some
         (
         Some
         {|
         l := lx;
         input := input item;
         destination :=
         equivocators_state_project IM eqv_item
         (destination item);
         output := output item
         |}, eqv_item)
is_singleton_state (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
    apply (not_equivocating_index_has_singleton_state _ Hbs _ finite_index equivocating); [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item 
  (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Hex: is_singleton_state 
  (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
→ ∀ s : composite_state (equivocator_IM IM),
    composite_valid 
      (equivocator_IM IM) 
      (l item) 
      (s, input item)
    → composite_transition 
        (equivocator_IM IM) 
        (l item) 
        (s, input item) =
      (destination item, output item)
      → ∃ Hex : 
         existing_equivocator_label
         (IM (projT1 (l item)))
         (projT2 (l item)),
         let lx :=
         existT 
         (projT1 (l item))
         (existing_equivocator_label_extract
         (IM (projT1 (l item)))
         (projT2 (l item)) Hex) in
         equivocators_transition_item_project IM
         eqv_item item =
         Some
         (
         Some
         {|
         l := lx;
         input := input item;
         destination :=
         equivocators_state_project IM eqv_item
         (destination item);
         output := output item
         |}, eqv_item)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM Hbs
     index_listing equivocating) (destination item)
    apply finite_valid_trace_last_pstate in Hitem.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: valid_state_prop XE
  (finite_trace_last
     (finite_trace_last is pre) [item])
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item 
  (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Hex: is_singleton_state 
  (IM (projT1 (l item)))
  (destination item (projT1 (l item)))
→ ∀ s : composite_state (equivocator_IM IM),
    composite_valid 
      (equivocator_IM IM) 
      (l item) 
      (s, input item)
    → composite_transition 
        (equivocator_IM IM) 
        (l item) 
        (s, input item) =
      (destination item, output item)
      → ∃ Hex : 
         existing_equivocator_label
         (IM (projT1 (l item)))
         (projT2 (l item)),
         let lx :=
         existT 
         (projT1 (l item))
         (existing_equivocator_label_extract
         (IM (projT1 (l item)))
         (projT2 (l item)) Hex) in
         equivocators_transition_item_project IM
         eqv_item item =
         Some
         (
         Some
         {|
         l := lx;
         input := input item;
         destination :=
         equivocators_state_project IM eqv_item
         (destination item);
         output := output item
         |}, eqv_item)
valid_state_prop
  (equivocators_fixed_equivocations_vlsm IM Hbs
     index_listing equivocating) (destination item) assumption.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre) [item]
Hsuf: finite_valid_trace_from XE 
  (destination item) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last (destination item) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [item] = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output item = Some m
Hno_equiv_item: projT1 (l item) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination item) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item 
  (destination item)
Hproper_fixed_item: eqv_item (projT1 (l item)) =
Existing 0
Hex: ∀ s : composite_state (equivocator_IM IM),
  composite_valid 
    (equivocator_IM IM) 
    (l item) 
    (s, input item)
  → composite_transition 
      (equivocator_IM IM) 
      (l item) 
      (s, input item) =
    (destination item, output item)
    → ∃ Hex : 
         existing_equivocator_label
         (IM (projT1 (l item)))
         (projT2 (l item)),
        let lx :=
         existT 
         (projT1 (l item))
         (existing_equivocator_label_extract
         (IM (projT1 (l item)))
         (projT2 (l item)) Hex) in
        equivocators_transition_item_project IM
         eqv_item item =
        Some
         (
         Some
         {|
         l := lx;
         input := input item;
         destination :=
         equivocators_state_project IM eqv_item
         (destination item);
         output := output item
         |}, eqv_item)
Exists (field_selector output m) itemXs
  destruct item as (l, iom, s, oom).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
l: label
iom: option message
s: state
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: finite_valid_trace_from XE
  (finite_trace_last is pre)
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}]
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |})) = 
Existing 0
Hex: ∀ s0 : composite_state (equivocator_IM IM),
  composite_valid 
    (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (s0,
    input
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |})
  → composite_transition 
      (equivocator_IM IM)
      (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})
      (s0,
      input
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) =
    (destination
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |},
    output
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |})
    → ∃ Hex : 
         existing_equivocator_label
         (IM
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
         (projT2
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})),
        let lx :=
         existT
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}))
         (existing_equivocator_label_extract
         (IM
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
         (projT2
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})) Hex) in
        equivocators_transition_item_project IM
         eqv_item
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |} =
        Some
         (
         Some
         {|
         l := lx;
         input :=
         input
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |};
         destination :=
         equivocators_state_project IM eqv_item
         (destination
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |});
         output :=
         output
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}
         |}, eqv_item)
Exists (field_selector output m) itemXs apply first_transition_valid in Hitem.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
l: label
iom: option message
s: state
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: input_valid_transition XE
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
  (finite_trace_last is pre,
  input
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |})
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |},
  output
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |})
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |})) = 
Existing 0
Hex: ∀ s0 : composite_state (equivocator_IM IM),
  composite_valid 
    (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (s0,
    input
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |})
  → composite_transition 
      (equivocator_IM IM)
      (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})
      (s0,
      input
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) =
    (destination
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |},
    output
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |})
    → ∃ Hex : 
         existing_equivocator_label
         (IM
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
         (projT2
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})),
        let lx :=
         existT
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}))
         (existing_equivocator_label_extract
         (IM
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
         (projT2
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})) Hex) in
        equivocators_transition_item_project IM
         eqv_item
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |} =
        Some
         (
         Some
         {|
         l := lx;
         input :=
         input
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |};
         destination :=
         equivocators_state_project IM eqv_item
         (destination
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |});
         output :=
         output
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}
         |}, eqv_item)
Exists (field_selector output m) itemXs simpl in Hitem.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
l: label
iom: option message
s: state
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hitem: input_valid_transition XE l
  (finite_trace_last is pre, iom) 
  (s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |})) = 
Existing 0
Hex: ∀ s0 : composite_state (equivocator_IM IM),
  composite_valid 
    (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (s0,
    input
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |})
  → composite_transition 
      (equivocator_IM IM)
      (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})
      (s0,
      input
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) =
    (destination
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |},
    output
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |})
    → ∃ Hex : 
         existing_equivocator_label
         (IM
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
         (projT2
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})),
        let lx :=
         existT
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}))
         (existing_equivocator_label_extract
         (IM
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
         (projT2
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})) Hex) in
        equivocators_transition_item_project IM
         eqv_item
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |} =
        Some
         (
         Some
         {|
         l := lx;
         input :=
         input
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |};
         destination :=
         equivocators_state_project IM eqv_item
         (destination
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |});
         output :=
         output
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}
         |}, eqv_item)
Exists (field_selector output m) itemXs
  destruct Hitem as [[Hs [Hiom [Hv Hc]]] Ht].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
l: label
iom: option message
s: state
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |})) = 
Existing 0
Hex: ∀ s0 : composite_state (equivocator_IM IM),
  composite_valid 
    (equivocator_IM IM)
    (VLSM.l
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |})
    (s0,
    input
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |})
  → composite_transition 
      (equivocator_IM IM)
      (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})
      (s0,
      input
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) =
    (destination
       {|
         l := l;
         input := iom;
         destination := s;
         output := oom
       |},
    output
      {|
        l := l;
        input := iom;
        destination := s;
        output := oom
      |})
    → ∃ Hex : 
         existing_equivocator_label
         (IM
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
         (projT2
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})),
        let lx :=
         existT
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}))
         (existing_equivocator_label_extract
         (IM
         (projT1
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})))
         (projT2
         (VLSM.l
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |})) Hex) in
        equivocators_transition_item_project IM
         eqv_item
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |} =
        Some
         (
         Some
         {|
         l := lx;
         input :=
         input
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |};
         destination :=
         equivocators_state_project IM eqv_item
         (destination
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |});
         output :=
         output
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |}
         |}, eqv_item)
Exists (field_selector output m) itemXs
  simpl in Hex.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
l: label
iom: option message
s: state
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |})) = 
Existing 0
Hex: ∀ s0 : composite_state (equivocator_IM IM),
  (let (i, li) := l in
   vvalid (equivocator_IM IM i) li (s0 i, iom))
  → (let (i, li) := l in
     let (si', om') :=
       vtransition 
         (equivocator_IM IM i) li 
         (s0 i, iom) in
     (state_update (equivocator_IM IM) s0 i si',
     om')) = 
    (s, oom)
    → ∃ Hex : 
         existing_equivocator_label
         (IM (projT1 l)) 
         (projT2 l),
        equivocators_transition_item_project IM
         eqv_item
         {|
         l := l;
         input := iom;
         destination := s;
         output := oom
         |} =
        Some
         (
         Some
         {|
         l :=
         existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex);
         input := iom;
         destination :=
         equivocators_state_project IM eqv_item
         s;
         output := oom
         |}, eqv_item)
Exists (field_selector output m) itemXs
  specialize (Hex _ Hv Ht) as [Hex' Hex].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
l: label
iom: option message
s: state
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |})) = 
Existing 0
Hex': existing_equivocator_label 
  (IM (projT1 l)) 
  (projT2 l)
Hex: equivocators_transition_item_project IM eqv_item
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l :=
         existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex');
       input := iom;
       destination :=
         equivocators_state_project IM eqv_item
         s;
       output := oom
     |}, eqv_item)
Exists (field_selector output m) itemXs simpl in Hex.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
l: label
iom: option message
s: state
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: equivocators_trace_project IM eqv_item
  [{|
     l := l;
     input := iom;
     destination := s;
     output := oom
   |}] = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |})) = 
Existing 0
Hex': existing_equivocator_label 
  (IM (projT1 l)) 
  (projT2 l)
Hex: equivocators_transition_item_project IM eqv_item
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l :=
         existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex');
       input := iom;
       destination :=
         equivocators_state_project IM eqv_item
         s;
       output := oom
     |}, eqv_item)
Exists (field_selector output m) itemXs
  simpl in Hpr_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
l: label
iom: option message
s: state
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hpr_item: match
  equivocators_transition_item_project IM
    eqv_item
    {|
      l := l;
      input := iom;
      destination := s;
      output := oom
    |}
with
| Some (Some item', odescriptor) =>
    Some ([item'], odescriptor)
| Some (None, odescriptor) =>
    Some ([], odescriptor)
| None => None
end = Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |})) = 
Existing 0
Hex': existing_equivocator_label 
  (IM (projT1 l)) 
  (projT2 l)
Hex: equivocators_transition_item_project IM eqv_item
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l :=
         existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex');
       input := iom;
       destination :=
         equivocators_state_project IM eqv_item
         s;
       output := oom
     |}, eqv_item)
Exists (field_selector output m) itemXs rewrite Hex in Hpr_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
l: label
iom: option message
s: state
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hex': existing_equivocator_label 
  (IM (projT1 l)) 
  (projT2 l)
Hpr_item: Some
  ([{|
      l :=
        existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex');
      input := iom;
      destination :=
        equivocators_state_project IM
         eqv_item s;
      output := oom
    |}], eqv_item) = 
Some (itemXs, eqv_pre)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |})) = 
Existing 0
Hex: equivocators_transition_item_project IM eqv_item
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l :=
         existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex');
       input := iom;
       destination :=
         equivocators_state_project IM eqv_item
         s;
       output := oom
     |}, eqv_item)
Exists (field_selector output m) itemXs
  inversion_clear Hpr_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
l: label
iom: option message
s: state
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hex': existing_equivocator_label 
  (IM (projT1 l)) 
  (projT2 l)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |})) = 
Existing 0
Hex: equivocators_transition_item_project IM eqv_item
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l :=
         existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex');
       input := iom;
       destination :=
         equivocators_state_project IM eqv_item
         s;
       output := oom
     |}, eqv_item)
Exists (field_selector output m)
  [{|
     l :=
       existT (projT1 l)
         (existing_equivocator_label_extract
            (IM (projT1 l)) (projT2 l) Hex');
     input := iom;
     destination :=
       equivocators_state_project IM eqv_pre s;
     output := oom
   |}]
  constructor.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: composite_state (equivocator_IM IM)
l: label
iom: option message
s: state
oom: option message
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Hs: valid_state_prop XE (finite_trace_last is pre)
Hiom: option_valid_message_prop XE iom
Hv: composite_valid 
  (equivocator_IM IM) l
  (finite_trace_last is pre, iom)
Hc: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing equivocating l
  (finite_trace_last is pre, iom)
Ht: transition l (finite_trace_last is pre, iom) =
(s, oom)
Hsuf: finite_valid_trace_from XE
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
m: message
final_descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors
  (finite_trace_last
     (destination
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |}) suf)
initial_descriptors: equivocator_descriptors IM
preX: list (composite_transition_item IM)
eqv_pre: equivocator_descriptors IM
itemXs, sufX: list (composite_transition_item IM)
eqv_item: equivocator_descriptors IM
Hpr_suf: equivocators_trace_project IM
  final_descriptors suf =
Some (sufX, eqv_item)
Hex': existing_equivocator_label 
  (IM (projT1 l)) 
  (projT2 l)
Hpr_pre: equivocators_trace_project IM eqv_pre pre =
Some (preX, initial_descriptors)
Houtput: output
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} = Some m
Hno_equiv_item: projT1
  (VLSM.l
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) ∉ equivocating
Hsufpre: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE)
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |}) suf
Hproper_item: proper_equivocator_descriptors IM
  eqv_item
  (destination
     {|
       l := l;
       input := iom;
       destination := s;
       output := oom
     |})
Hproper_fixed_item: eqv_item
  (projT1
     (VLSM.l
        {|
         l := l;
         input := iom;
         destination := s;
         output := oom
        |})) = 
Existing 0
Hex: equivocators_transition_item_project IM eqv_item
  {|
    l := l;
    input := iom;
    destination := s;
    output := oom
  |} =
Some
  (Some
     {|
       l :=
         existT 
         (projT1 l)
         (existing_equivocator_label_extract
         (IM (projT1 l)) 
         (projT2 l) Hex');
       input := iom;
       destination :=
         equivocators_state_project IM eqv_item
         s;
       output := oom
     |}, eqv_item)
field_selector output m
  {|
    l :=
      existT (projT1 l)
        (existing_equivocator_label_extract
           (IM (projT1 l)) (projT2 l) Hex');
    input := iom;
    destination :=
      equivocators_state_project IM eqv_pre s;
    output := oom
  |} assumption.
Qed.

As a consequence of the equivocator_vlsm_trace_project_reflect_non_equivocating lemma, if a message emmited by a trace cannot be observed in a projection of the trace's final state, then it must be that it was emitted by one of the nodes allowed to equivocate.

Lemma projection_has_not_been_observed_is_equivocating
  (is: _composite_state (equivocator_IM IM))
  (tr: list (composite_transition_item (equivocator_IM IM)))
  (Htr: finite_valid_trace XE is tr)
  (s := @finite_trace_last _ (composite_type (equivocator_IM IM)) is tr)
  (descriptors: equivocator_descriptors IM)
  (Hproper: proper_fixed_equivocator_descriptors descriptors s)
  (sX := equivocators_state_project IM descriptors s)
  (m: message)
  (Hno: ~ composite_has_been_observed IM Hbo sX m)
  : forall item : composite_transition_item (equivocator_IM IM),
      item ∈ tr -> output item = Some m -> (projT1 (l item)) ∈ equivocating.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
Hno: ¬ composite_has_been_observed IM Hbo sX m
∀ item : composite_transition_item (equivocator_IM IM),
  item ∈ tr
  → output item = Some m
    → projT1 (l item) ∈ equivocating
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
Hno: ¬ composite_has_been_observed IM Hbo sX m
∀ item : composite_transition_item (equivocator_IM IM),
  item ∈ tr
  → output item = Some m
    → projT1 (l item) ∈ equivocating
  destruct (free_equivocators_valid_trace_project descriptors is tr Hproper Htr)
    as [trX [initial_descriptors [_ [Htr_project [Hlast_state HtrX]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
Hno: ¬ composite_has_been_observed IM Hbo sX m
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
∀ item : composite_transition_item (equivocator_IM IM),
  item ∈ tr
  → output item = Some m
    → projT1 (l item) ∈ equivocating
  intros item Hitem Houtput.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
Hno: ¬ composite_has_been_observed IM Hbo sX m
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
projT1 (l item) ∈ equivocating
  destruct (decide ((projT1 (l item)) ∈ equivocating)); [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
Hno: ¬ composite_has_been_observed IM Hbo sX m
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
projT1 (l item) ∈ equivocating
  elim Hno.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
Hno: ¬ composite_has_been_observed IM Hbo sX m
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
composite_has_been_observed IM Hbo sX m clear Hno.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
composite_has_been_observed IM Hbo sX m
  apply composite_has_been_observed_sent_received_iff.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
composite_has_been_sent IM Hbs sX m
∨ composite_has_been_received IM Hbr sX m
  left.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
composite_has_been_sent IM Hbs sX m
  assert (HtrX_free : finite_valid_trace (pre_loaded_with_all_messages_vlsm Free) (equivocators_state_project IM initial_descriptors is) trX).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM initial_descriptors
     is) trX
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
HtrX_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
composite_has_been_sent IM Hbs sX m
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM initial_descriptors
     is) trX
    revert HtrX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM initial_descriptors
     is) trX
→ finite_valid_trace
    (pre_loaded_with_all_messages_vlsm Free)
    (equivocators_state_project IM initial_descriptors
       is) trX clear.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
is: _composite_state (equivocator_IM IM)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
finite_valid_trace (free_composite_vlsm IM)
  (equivocators_state_project IM initial_descriptors
     is) trX
→ finite_valid_trace
    (pre_loaded_with_all_messages_vlsm Free)
    (equivocators_state_project IM initial_descriptors
       is) trX apply VLSM_incl_finite_valid_trace.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Free:= free_composite_vlsm IM: VLSM message
is: _composite_state (equivocator_IM IM)
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
VLSM_incl_part
  (composite_vlsm_machine IM (free_constraint IM))
  (pre_loaded_with_all_messages_vlsm_machine Free)
    apply vlsm_incl_pre_loaded_with_all_messages_vlsm.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
HtrX_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
composite_has_been_sent IM Hbs sX m
  specialize (finite_valid_trace_last_pstate _ _ _ (proj1 HtrX_free)) as Hfree_lst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
s:= finite_trace_last is tr: state
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors s
sX:= equivocators_state_project IM descriptors s: vstate (free_composite_vlsm IM)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
HtrX_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
composite_has_been_sent IM Hbs sX m
  subst sX s.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
HtrX_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
composite_has_been_sent IM Hbs
  (equivocators_state_project IM descriptors
     (finite_trace_last is tr)) m
  simpl in *.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
HtrX_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
composite_has_been_sent IM Hbs
  (equivocators_state_project IM descriptors
     (finite_trace_last is tr)) m
  rewrite Hlast_state.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
HtrX_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
composite_has_been_sent IM Hbs
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  apply (composite_proper_sent IM finite_index Hbs)
  ; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
HtrX_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
selected_message_exists_in_all_preloaded_traces
  (free_composite_vlsm IM) (field_selector output)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  apply has_been_sent_consistency; [assumption| assumption |].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
HtrX_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
selected_message_exists_in_some_preloaded_traces
  (free_composite_vlsm IM) (field_selector output)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX) m
  exists (equivocators_state_project IM initial_descriptors is), trX, (valid_trace_add_default_last HtrX_free).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
HtrX: finite_valid_trace 
  (free_composite_vlsm IM)
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
HtrX_free: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm Free)
  (equivocators_state_project IM
     initial_descriptors is) trX
Hfree_lst: valid_state_prop
  (pre_loaded_with_all_messages_vlsm Free)
  (finite_trace_last
     (equivocators_state_project IM
        initial_descriptors is) trX)
trace_has_message (field_selector output) m trX
  clear HtrX HtrX_free Hfree_lst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
is: _composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace XE is tr
descriptors: equivocator_descriptors IM
Hproper: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
m: message
trX: list (composite_transition_item IM)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  descriptors tr =
Some (trX, initial_descriptors)
Hlast_state: equivocators_state_project IM
  descriptors 
  (finite_trace_last is tr) =
finite_trace_last
  (equivocators_state_project IM
     initial_descriptors is) trX
item: composite_transition_item (equivocator_IM IM)
Hitem: item ∈ tr
Houtput: output item = Some m
n: projT1 (l item) ∉ equivocating
trace_has_message (field_selector output) m trX
  apply equivocator_vlsm_trace_project_reflect_non_equivocating with is tr descriptors initial_descriptors item
  ; assumption.
Qed.

The next two lemmas are rather technical, but their basic meaning is that if we take the composition of a subset of equivocators, it doesn't matter the way we do it (either first obtain the indexed subset of nodes and then transform that into equivocators and take their composition, or we first start with the indexed full set of equivocators and then select a subset of them and take their composition).

Lemma pre_loaded_equivocators_composition_sub_projection_commute
  (seed1 : message -> Prop)
  (seed2 : message -> Prop)
  (Hseed12 : forall m, seed1 m -> seed2 m)
  : VLSM_incl
    (composite_no_equivocation_vlsm_with_pre_loaded
      (sub_IM (equivocator_IM IM) equivocating)
      (free_constraint _)
      (sub_has_been_sent_capabilities (equivocator_IM IM)
        equivocating (equivocator_Hbs IM Hbs))
      seed1)
    (composite_no_equivocation_vlsm_with_pre_loaded
      (equivocator_IM (sub_IM IM equivocating))
      (free_constraint _)
      (equivocator_Hbs (sub_IM IM equivocating)
        (sub_has_been_sent_capabilities IM equivocating Hbs))
      seed2).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
VLSM_incl
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs)) seed1)
  (composite_no_equivocation_vlsm_with_pre_loaded
     (equivocator_IM (sub_IM IM equivocating))
     (free_constraint
        (equivocator_IM (sub_IM IM equivocating)))
     (equivocator_Hbs (sub_IM IM equivocating)
        (sub_has_been_sent_capabilities IM
           equivocating Hbs)) seed2)
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
VLSM_incl
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs)) seed1)
  (composite_no_equivocation_vlsm_with_pre_loaded
     (equivocator_IM (sub_IM IM equivocating))
     (free_constraint
        (equivocator_IM (sub_IM IM equivocating)))
     (equivocator_Hbs (sub_IM IM equivocating)
        (sub_has_been_sent_capabilities IM
           equivocating Hbs)) seed2)
  apply basic_VLSM_incl
  ; intro; intros; [assumption|..].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) l 
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HmX: vinitial_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) m
valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
           (projT2
              (composite_vlsm
                 (equivocator_IM
                    (sub_IM IM equivocating))
                 (no_equivocations_additional_constraint_with_pre_loaded
                    (equivocator_IM
                       (sub_IM IM equivocating))
                    (free_constraint
                       (equivocator_IM
                          (sub_IM IM equivocating)))
                    (equivocator_Hbs
                       (sub_IM IM equivocating)
                       (sub_has_been_sent_capabilities
                          IM equivocating Hbs)) seed2))))
        seed2)) m
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) l 
  (s, om)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  om
vvalid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs 
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2)) 
  (id l) (id s, om)
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
s': state
om': option message
H: input_valid_transition
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1)))) seed1))
  l (s, om) 
  (s', om')
vtransition
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs 
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2)) 
  (id l) (id s, om) = 
(id s', om')
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) l 
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HmX: vinitial_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) m
valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
           (projT2
              (composite_vlsm
                 (equivocator_IM
                    (sub_IM IM equivocating))
                 (no_equivocations_additional_constraint_with_pre_loaded
                    (equivocator_IM
                       (sub_IM IM equivocating))
                    (free_constraint
                       (equivocator_IM
                          (sub_IM IM equivocating)))
                    (equivocator_Hbs
                       (sub_IM IM equivocating)
                       (sub_has_been_sent_capabilities
                          IM equivocating Hbs)) seed2))))
        seed2)) m apply initial_message_is_valid.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) l 
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HmX: vinitial_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) m
initial_message_prop m
    simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) l 
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HmX: vinitial_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) m
composite_initial_message_prop
  (equivocator_IM (sub_IM IM equivocating)) m
∨ seed2 m
    destruct HmX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) l 
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
H: initial_message_prop m
composite_initial_message_prop
  (equivocator_IM (sub_IM IM equivocating)) m
∨ seed2 m
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) l 
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
H: seed1 m
composite_initial_message_prop
  (equivocator_IM (sub_IM IM equivocating)) m
∨ seed2 m
    +message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) l 
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
H: initial_message_prop m
composite_initial_message_prop
  (equivocator_IM (sub_IM IM equivocating)) m
∨ seed2 m left; assumption.
    +message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) l 
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
H: seed1 m
composite_initial_message_prop
  (equivocator_IM (sub_IM IM equivocating)) m
∨ seed2 m right.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) l 
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
H: seed1 m
seed2 m by apply Hseed12.
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) l 
  (s, om)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  om
vvalid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
           (projT2
              (composite_vlsm
                 (equivocator_IM
                    (sub_IM IM equivocating))
                 (no_equivocations_additional_constraint_with_pre_loaded
                    (equivocator_IM
                       (sub_IM IM equivocating))
                    (free_constraint
                       (equivocator_IM
                          (sub_IM IM equivocating)))
                    (equivocator_Hbs
                       (sub_IM IM equivocating)
                       (sub_has_been_sent_capabilities
                          IM equivocating Hbs)) seed2))))
        seed2)) (id l) (id s, om) destruct Hv as [Hs [_ [Hv Hc]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, om)
Hc: no_equivocations_additional_constraint_with_pre_loaded
  (sub_IM (equivocator_IM IM) equivocating)
  (free_constraint
     (sub_IM (equivocator_IM IM) equivocating))
  (sub_has_been_sent_capabilities
     (equivocator_IM IM) equivocating
     (equivocator_Hbs IM Hbs)) seed1 l 
  (s, om)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  om
vvalid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
           (projT2
              (composite_vlsm
                 (equivocator_IM
                    (sub_IM IM equivocating))
                 (no_equivocations_additional_constraint_with_pre_loaded
                    (equivocator_IM
                       (sub_IM IM equivocating))
                    (free_constraint
                       (equivocator_IM
                          (sub_IM IM equivocating)))
                    (equivocator_Hbs
                       (sub_IM IM equivocating)
                       (sub_has_been_sent_capabilities
                          IM equivocating Hbs)) seed2))))
        seed2)) (id l) (id s, om)
    split; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, om)
Hc: no_equivocations_additional_constraint_with_pre_loaded
  (sub_IM (equivocator_IM IM) equivocating)
  (free_constraint
     (sub_IM (equivocator_IM IM) equivocating))
  (sub_has_been_sent_capabilities
     (equivocator_IM IM) equivocating
     (equivocator_Hbs IM Hbs)) seed1 l 
  (s, om)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  om
no_equivocations_additional_constraint_with_pre_loaded
  (equivocator_IM (sub_IM IM equivocating))
  (free_constraint
     (equivocator_IM (sub_IM IM equivocating)))
  (equivocator_Hbs (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM equivocating
        Hbs)) seed2 (id l) (id s, om)
    destruct Hc as [Hc _].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, om)
Hc: composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities
     (equivocator_IM IM) equivocating
     (equivocator_Hbs IM Hbs)) seed1 l 
  (s, om)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  om
no_equivocations_additional_constraint_with_pre_loaded
  (equivocator_IM (sub_IM IM equivocating))
  (free_constraint
     (equivocator_IM (sub_IM IM equivocating)))
  (equivocator_Hbs (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM equivocating
        Hbs)) seed2 (id l) (id s, om)
    split; [|exact I].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, om)
Hc: composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities
     (equivocator_IM IM) equivocating
     (equivocator_Hbs IM Hbs)) seed1 l 
  (s, om)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  om
composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM equivocating
        Hbs)) seed2 (id l) (id s, om)
    destruct om; [| exact I].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, Some m)
Hc: composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities
     (equivocator_IM IM) equivocating
     (equivocator_Hbs IM Hbs)) seed1 l
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (Some m)
composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM equivocating
        Hbs)) seed2 (id l) (id s, Some m)
    simpl in Hc.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, Some m)
Hc: composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities
     (equivocator_IM IM) equivocating
     (equivocator_Hbs IM Hbs)) seed1 l
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (Some m)
composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM equivocating
        Hbs)) seed2 (id l) (id s, Some m) simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, Some m)
Hc: composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities
     (equivocator_IM IM) equivocating
     (equivocator_Hbs IM Hbs)) seed1 l
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (Some m)
composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM equivocating
        Hbs)) seed2 l (s, Some m)
    destruct Hc as [Hc | Hc ]
    ; [| right; apply Hseed12; assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, Some m)
Hc: composite_has_been_sent
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities
     (equivocator_IM IM) equivocating
     (equivocator_Hbs IM Hbs)) 
  (s, Some m).1 m
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (Some m)
composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM equivocating
        Hbs)) seed2 l (s, Some m)
    left.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, Some m)
Hc: composite_has_been_sent
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities
     (equivocator_IM IM) equivocating
     (equivocator_Hbs IM Hbs)) 
  (s, Some m).1 m
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (Some m)
composite_has_been_sent
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM equivocating
        Hbs)) (s, Some m).1 m
    destruct Hc as [subi Hibs].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, Some m)
subi: sub_index equivocating
Hibs: has_been_sent
  (sub_IM (equivocator_IM IM) equivocating subi)
  ((s, Some m).1 subi) m
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (Some m)
composite_has_been_sent
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM equivocating
        Hbs)) (s, Some m).1 m
    exists subi.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, Some m)
subi: sub_index equivocating
Hibs: has_been_sent
  (sub_IM (equivocator_IM IM) equivocating subi)
  ((s, Some m).1 subi) m
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (Some m)
has_been_sent
  (equivocator_IM (sub_IM IM equivocating) subi)
  ((s, Some m).1 subi) m revert Hibs.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, Some m)
subi: sub_index equivocating
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (Some m)
has_been_sent
  (sub_IM (equivocator_IM IM) equivocating subi)
  ((s, Some m).1 subi) m
→ has_been_sent
    (equivocator_IM (sub_IM IM equivocating) subi)
    ((s, Some m).1 subi) m
    apply has_been_sent_irrelevance.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, Some m)
subi: sub_index equivocating
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (Some m)
valid_state_prop
  (pre_loaded_with_all_messages_vlsm
     (equivocator_IM (sub_IM IM equivocating) subi))
  ((s, Some m).1 subi)
    simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, Some m)
subi: sub_index equivocating
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (Some m)
valid_state_prop
  (pre_loaded_with_all_messages_vlsm
     (equivocator_IM (sub_IM IM equivocating) subi))
  (s subi)
    apply (preloaded_valid_state_projection (equivocator_IM (sub_IM IM equivocating)) subi).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, Some m)
subi: sub_index equivocating
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (Some m)
valid_state_prop
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM (sub_IM IM equivocating)))) s
    revert Hs.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, Some m)
subi: sub_index equivocating
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (Some m)
valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
           (projT2
              (composite_vlsm
                 (sub_IM (equivocator_IM IM)
                    equivocating)
                 (no_equivocations_additional_constraint_with_pre_loaded
                    (sub_IM (equivocator_IM IM)
                       equivocating)
                    (free_constraint
                       (sub_IM (equivocator_IM IM)
                          equivocating))
                    (sub_has_been_sent_capabilities
                       (equivocator_IM IM)
                       equivocating
                       (equivocator_Hbs IM Hbs)) seed1))))
        seed1)) s
→ valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (free_composite_vlsm
          (equivocator_IM (sub_IM IM equivocating))))
    s
    apply VLSM_incl_valid_state.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: composite_valid
  (sub_IM (equivocator_IM IM) equivocating) l
  (s, Some m)
subi: sub_index equivocating
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed2)))) seed2))
  (Some m)
VLSM_incl_part
  (VLSMClass_pre_loaded_with_messages
     (projT2
        (projT2
           (composite_vlsm
              (sub_IM (equivocator_IM IM) equivocating)
              (no_equivocations_additional_constraint_with_pre_loaded
                 (sub_IM (equivocator_IM IM)
                    equivocating)
                 (free_constraint
                    (sub_IM (equivocator_IM IM)
                       equivocating))
                 (sub_has_been_sent_capabilities
                    (equivocator_IM IM) equivocating
                    (equivocator_Hbs IM Hbs)) seed1))))
     seed1)
  (pre_loaded_with_all_messages_vlsm_machine
     (free_composite_vlsm
        (equivocator_IM (sub_IM IM equivocating))))
    apply composite_pre_loaded_vlsm_incl_pre_loaded_with_all_messages.
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
s': state
om': option message
H: input_valid_transition
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed1)))) seed1))
  l (s, om) 
  (s', om')
vtransition
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
           (projT2
              (composite_vlsm
                 (equivocator_IM
                    (sub_IM IM equivocating))
                 (no_equivocations_additional_constraint_with_pre_loaded
                    (equivocator_IM
                       (sub_IM IM equivocating))
                    (free_constraint
                       (equivocator_IM
                          (sub_IM IM equivocating)))
                    (equivocator_Hbs
                       (sub_IM IM equivocating)
                       (sub_has_been_sent_capabilities
                          IM equivocating Hbs)) seed2))))
        seed2)) (id l) (id s, om) = (id s', om') apply H.
Qed.

Lemma pre_loaded_equivocators_composition_sub_projection_commute_inv
  (seed1 : message -> Prop)
  (seed2 : message -> Prop)
  (Hseed12 : forall m, seed1 m -> seed2 m)
  : VLSM_incl
    (composite_no_equivocation_vlsm_with_pre_loaded
      (equivocator_IM (sub_IM IM equivocating))
      (free_constraint _)
      (equivocator_Hbs (sub_IM IM equivocating)
        (sub_has_been_sent_capabilities IM equivocating Hbs))
      seed1)
    (composite_no_equivocation_vlsm_with_pre_loaded
      (sub_IM (equivocator_IM IM) equivocating)
      (free_constraint _)
      (sub_has_been_sent_capabilities (equivocator_IM IM)
        equivocating (equivocator_Hbs IM Hbs))
      seed2).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
VLSM_incl
  (composite_no_equivocation_vlsm_with_pre_loaded
     (equivocator_IM (sub_IM IM equivocating))
     (free_constraint
        (equivocator_IM (sub_IM IM equivocating)))
     (equivocator_Hbs (sub_IM IM equivocating)
        (sub_has_been_sent_capabilities IM
           equivocating Hbs)) seed1)
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs)) seed2)
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
VLSM_incl
  (composite_no_equivocation_vlsm_with_pre_loaded
     (equivocator_IM (sub_IM IM equivocating))
     (free_constraint
        (equivocator_IM (sub_IM IM equivocating)))
     (equivocator_Hbs (sub_IM IM equivocating)
        (sub_has_been_sent_capabilities IM
           equivocating Hbs)) seed1)
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs)) seed2)
  apply basic_VLSM_incl
  ; intro; intros; [assumption|..].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) l
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HmX: vinitial_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) m
valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
           (projT2
              (composite_vlsm
                 (sub_IM (equivocator_IM IM)
                    equivocating)
                 (no_equivocations_additional_constraint_with_pre_loaded
                    (sub_IM (equivocator_IM IM)
                       equivocating)
                    (free_constraint
                       (sub_IM (equivocator_IM IM)
                          equivocating))
                    (sub_has_been_sent_capabilities
                       (equivocator_IM IM)
                       equivocating
                       (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) m
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) l
  (s, om)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) om
vvalid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2)))) seed2))
  (id l) (id s, om)
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
s': state
om': option message
H: input_valid_transition
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs 
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) l
  (s, om) 
  (s', om')
vtransition
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2)))) seed2))
  (id l) (id s, om) = 
(id s', om')
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) l
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HmX: vinitial_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) m
valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
           (projT2
              (composite_vlsm
                 (sub_IM (equivocator_IM IM)
                    equivocating)
                 (no_equivocations_additional_constraint_with_pre_loaded
                    (sub_IM (equivocator_IM IM)
                       equivocating)
                    (free_constraint
                       (sub_IM (equivocator_IM IM)
                          equivocating))
                    (sub_has_been_sent_capabilities
                       (equivocator_IM IM)
                       equivocating
                       (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) m apply initial_message_is_valid.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) l
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HmX: vinitial_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) m
initial_message_prop m
    simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) l
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HmX: vinitial_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM
         (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM
         (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM
         (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) m
composite_initial_message_prop
  (sub_IM (equivocator_IM IM) equivocating) m
∨ seed2 m
    destruct HmX; [left; assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) l
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
H: seed1 m
composite_initial_message_prop
  (sub_IM (equivocator_IM IM) equivocating) m
∨ seed2 m
    right.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) l
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
H: seed1 m
seed2 m
    apply Hseed12; assumption.
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
Hv: input_valid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) l
  (s, om)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) om
vvalid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
           (projT2
              (composite_vlsm
                 (sub_IM (equivocator_IM IM)
                    equivocating)
                 (no_equivocations_additional_constraint_with_pre_loaded
                    (sub_IM (equivocator_IM IM)
                       equivocating)
                    (free_constraint
                       (sub_IM (equivocator_IM IM)
                          equivocating))
                    (sub_has_been_sent_capabilities
                       (equivocator_IM IM)
                       equivocating
                       (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) (id l) (id s, om) destruct Hv as [Hs [_ [Hv [Hc _]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, om)
Hc: composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs 
     (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed1 l 
  (s, om)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) om
vvalid
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
           (projT2
              (composite_vlsm
                 (sub_IM (equivocator_IM IM)
                    equivocating)
                 (no_equivocations_additional_constraint_with_pre_loaded
                    (sub_IM (equivocator_IM IM)
                       equivocating)
                    (free_constraint
                       (sub_IM (equivocator_IM IM)
                          equivocating))
                    (sub_has_been_sent_capabilities
                       (equivocator_IM IM)
                       equivocating
                       (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) (id l) (id s, om)
    split; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, om)
Hc: composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs 
     (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed1 l 
  (s, om)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) om
no_equivocations_additional_constraint_with_pre_loaded
  (sub_IM (equivocator_IM IM) equivocating)
  (free_constraint
     (sub_IM (equivocator_IM IM) equivocating))
  (sub_has_been_sent_capabilities (equivocator_IM IM)
     equivocating (equivocator_Hbs IM Hbs)) seed2
  (id l) (id s, om)
    split; [|exact I].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, om)
Hc: composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs 
     (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed1 l 
  (s, om)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) om
composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities (equivocator_IM IM)
     equivocating (equivocator_Hbs IM Hbs)) seed2
  (id l) (id s, om)
    destruct om; [| exact I].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, Some m)
Hc: composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs 
     (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed1 l 
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (Some m)
composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities (equivocator_IM IM)
     equivocating (equivocator_Hbs IM Hbs)) seed2
  (id l) (id s, Some m)
    simpl in Hc.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, Some m)
Hc: composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs 
     (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed1 l 
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (Some m)
composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities (equivocator_IM IM)
     equivocating (equivocator_Hbs IM Hbs)) seed2
  (id l) (id s, Some m) simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, Some m)
Hc: composite_no_equivocations_except_from
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs 
     (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) seed1 l 
  (s, Some m)
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (Some m)
composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities (equivocator_IM IM)
     equivocating (equivocator_Hbs IM Hbs)) seed2 l
  (s, Some m)
    destruct Hc as [Hc | Hc]
    ; [| right; apply Hseed12; assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, Some m)
Hc: composite_has_been_sent
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs 
     (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) 
  (s, Some m).1 m
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (Some m)
composite_no_equivocations_except_from
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities (equivocator_IM IM)
     equivocating (equivocator_Hbs IM Hbs)) seed2 l
  (s, Some m)
    left.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, Some m)
Hc: composite_has_been_sent
  (equivocator_IM (sub_IM IM equivocating))
  (equivocator_Hbs 
     (sub_IM IM equivocating)
     (sub_has_been_sent_capabilities IM
        equivocating Hbs)) 
  (s, Some m).1 m
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (Some m)
composite_has_been_sent
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities (equivocator_IM IM)
     equivocating (equivocator_Hbs IM Hbs))
  (s, Some m).1 m
    destruct Hc as [subi Hibs].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, Some m)
subi: sub_index equivocating
Hibs: has_been_sent
  (equivocator_IM (sub_IM IM equivocating) subi)
  ((s, Some m).1 subi) m
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (Some m)
composite_has_been_sent
  (sub_IM (equivocator_IM IM) equivocating)
  (sub_has_been_sent_capabilities (equivocator_IM IM)
     equivocating (equivocator_Hbs IM Hbs))
  (s, Some m).1 m
    exists subi.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, Some m)
subi: sub_index equivocating
Hibs: has_been_sent
  (equivocator_IM (sub_IM IM equivocating) subi)
  ((s, Some m).1 subi) m
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (Some m)
has_been_sent
  (sub_IM (equivocator_IM IM) equivocating subi)
  ((s, Some m).1 subi) m revert Hibs.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, Some m)
subi: sub_index equivocating
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (Some m)
has_been_sent
  (equivocator_IM (sub_IM IM equivocating) subi)
  ((s, Some m).1 subi) m
→ has_been_sent
    (sub_IM (equivocator_IM IM) equivocating subi)
    ((s, Some m).1 subi) m
    apply has_been_sent_irrelevance.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, Some m)
subi: sub_index equivocating
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (Some m)
valid_state_prop
  (pre_loaded_with_all_messages_vlsm
     (sub_IM (equivocator_IM IM) equivocating subi))
  ((s, Some m).1 subi)
    simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, Some m)
subi: sub_index equivocating
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (Some m)
valid_state_prop
  (pre_loaded_with_all_messages_vlsm
     (sub_IM (equivocator_IM IM) equivocating subi))
  (s subi)
    apply (preloaded_valid_state_projection (equivocator_IM (sub_IM IM equivocating)) subi).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hs: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) s
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, Some m)
subi: sub_index equivocating
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (Some m)
valid_state_prop
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm
        (equivocator_IM (sub_IM IM equivocating)))) s
    revert Hs.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, Some m)
subi: sub_index equivocating
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (Some m)
valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
           (projT2
              (composite_vlsm
                 (equivocator_IM
                    (sub_IM IM equivocating))
                 (no_equivocations_additional_constraint_with_pre_loaded
                    (equivocator_IM
                       (sub_IM IM equivocating))
                    (free_constraint
                       (equivocator_IM
                          (sub_IM IM equivocating)))
                    (equivocator_Hbs
                       (sub_IM IM equivocating)
                       (sub_has_been_sent_capabilities
                          IM equivocating Hbs)) seed1))))
        seed1)) s
→ valid_state_prop
    (pre_loaded_with_all_messages_vlsm
       (free_composite_vlsm
          (equivocator_IM (sub_IM IM equivocating))))
    s
    apply VLSM_incl_valid_state.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
m: message
Hv: composite_valid
  (equivocator_IM (sub_IM IM equivocating)) l
  (s, Some m)
subi: sub_index equivocating
HsY: valid_state_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (id s)
HomY: option_valid_message_prop
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (no_equivocations_additional_constraint_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) 
  (Some m)
VLSM_incl_part
  (VLSMClass_pre_loaded_with_messages
     (projT2
        (projT2
           (composite_vlsm
              (equivocator_IM (sub_IM IM equivocating))
              (no_equivocations_additional_constraint_with_pre_loaded
                 (equivocator_IM
                    (sub_IM IM equivocating))
                 (free_constraint
                    (equivocator_IM
                       (sub_IM IM equivocating)))
                 (equivocator_Hbs
                    (sub_IM IM equivocating)
                    (sub_has_been_sent_capabilities IM
                       equivocating Hbs)) seed1))))
     seed1)
  (pre_loaded_with_all_messages_vlsm_machine
     (free_composite_vlsm
        (equivocator_IM (sub_IM IM equivocating))))
    apply composite_pre_loaded_vlsm_incl_pre_loaded_with_all_messages.
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
seed1, seed2: message → Prop
Hseed12: ∀ m : message, seed1 m → seed2 m
l: label
s: state
om: option message
s': state
om': option message
H: input_valid_transition
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
         (projT2
         (composite_vlsm
         (equivocator_IM (sub_IM IM equivocating))
         (no_equivocations_additional_constraint_with_pre_loaded
         (equivocator_IM (sub_IM IM equivocating))
         (free_constraint
         (equivocator_IM (sub_IM IM equivocating)))
         (equivocator_Hbs 
         (sub_IM IM equivocating)
         (sub_has_been_sent_capabilities IM
         equivocating Hbs)) seed1)))) seed1)) l
  (s, om) 
  (s', om')
vtransition
  (mk_vlsm
     (VLSMClass_pre_loaded_with_messages
        (projT2
           (projT2
              (composite_vlsm
                 (sub_IM (equivocator_IM IM)
                    equivocating)
                 (no_equivocations_additional_constraint_with_pre_loaded
                    (sub_IM (equivocator_IM IM)
                       equivocating)
                    (free_constraint
                       (sub_IM (equivocator_IM IM)
                          equivocating))
                    (sub_has_been_sent_capabilities
                       (equivocator_IM IM)
                       equivocating
                       (equivocator_Hbs IM Hbs)) seed2))))
        seed2)) (id l) (id s, om) = (id s', om') apply H.
Qed.

The intermediary results above allow us to prove that the fixed_equivocation_constraint has the constraint_has_been_sent_property.

The core of this result is proving that given a valid_state s of the composition of equivocators with no message equivocation and fixed state equivocation, a message which has_been_sent for that state but not has_been_observed for a projection of that state sx, can nevertheless be generated by the composition of the nodes allowed to equivocate, pre-loaded with the messages observed in the state sx.

To prove that, we consider a trace witness for the mesage having been sent, we use projection_has_not_been_observed_is_equivocating to derive that it must have been sent by one of the machines allowed to equivocate, from this we derive that it can be sent by the restriction of the composition of equivocators to just the equivocating nodes, pre-loaded with the messages observed in the projection, then we use the seeded_equivocators_valid_trace_project result to reach our conclusion.

Lemma fixed_equivocation_constraint_has_constraint_has_been_sent_prop
  : constraint_has_been_sent_prop
    (fixed_equivocation_constraint IM Hbs Hbr equivocating).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
constraint_has_been_sent_prop
  (fixed_equivocation_constraint IM Hbs Hbr
     equivocating)
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
constraint_has_been_sent_prop
  (fixed_equivocation_constraint IM Hbs Hbr
     equivocating)
  unfold constraint_has_been_sent_prop.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
∀ s : composite_state (equivocator_IM IM),
  valid_state_prop XE s
  → ∀ descriptors : equivocator_descriptors IM,
      proper_fixed_equivocator_descriptors descriptors
        s
      → ∀ m : message,
          has_been_sent FreeE s m
          → ∀ l : composite_label IM,
              fixed_equivocation_constraint IM Hbs Hbr
                equivocating l
                (equivocators_state_project IM
                   descriptors s, Some m) intros.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
l: composite_label IM
fixed_equivocation_constraint IM Hbs Hbr equivocating
  l
  (equivocators_state_project IM descriptors s,
  Some m)
  remember (equivocators_state_project _ _ _) as sX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
l: composite_label IM
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
fixed_equivocation_constraint IM Hbs Hbr equivocating
  l (sX, Some m)
  destruct (composite_has_been_observed_dec IM finite_index Hbo sX m)
  ; [left; assumption|right].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
l: composite_label IM
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  clear l.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  unfold no_additional_equivocations in n.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: valid_state_prop XE s
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m

  (* Phase I: exhibiting a [valid_trace] ending in tr s and sending m *)

  apply valid_state_has_trace in Hs.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
Hs: ∃ (is : state) (tr : list transition_item),
  finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  destruct Hs as [is [tr Htr]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  (* subst s *)

  assert (Htr'pre : finite_valid_trace_init_to (pre_loaded_with_all_messages_vlsm FreeE) is s tr).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE) is s tr
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE) is s tr revert Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
finite_valid_trace_init_to XE is s tr
→ finite_valid_trace_init_to
    (pre_loaded_with_all_messages_vlsm FreeE) is s tr apply VLSM_incl_finite_valid_trace_init_to.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
VLSM_incl_part
  (composite_vlsm_machine (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint IM
        Hbs index_listing equivocating))
  (pre_loaded_with_all_messages_vlsm_machine FreeE)
    apply VLSM_incl_trans with (machine FreeE).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
VLSM_incl
  (mk_vlsm
     {|
       transition :=
         composite_transition (equivocator_IM IM);
       valid :=
         constrained_composite_valid
           (equivocator_IM IM)
           (equivocators_fixed_equivocations_constraint
              IM Hbs index_listing equivocating)
     |}) (mk_vlsm (machine FreeE))
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
VLSM_incl 
  (mk_vlsm (machine FreeE))
  (mk_vlsm
     {|
       transition := vtransition FreeE;
       valid := vvalid FreeE
     |})
    -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
VLSM_incl
  (mk_vlsm
     {|
       transition :=
         composite_transition (equivocator_IM IM);
       valid :=
         constrained_composite_valid
           (equivocator_IM IM)
           (equivocators_fixed_equivocations_constraint
              IM Hbs index_listing equivocating)
     |}) (mk_vlsm (machine FreeE)) apply
      (constraint_free_incl (equivocator_IM IM) (equivocators_fixed_equivocations_constraint IM Hbs
        index_listing equivocating)).
    -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
VLSM_incl (mk_vlsm (machine FreeE))
  (mk_vlsm
     {|
       transition := vtransition FreeE;
       valid := vvalid FreeE
     |}) apply vlsm_incl_pre_loaded_with_all_messages_vlsm.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  assert (Hplst : valid_state_prop (pre_loaded_with_all_messages_vlsm FreeE) s).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE) s
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hplst: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE) s
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE) s apply valid_trace_last_pstate in Htr'pre; assumption. }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: has_been_sent FreeE s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hplst: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE) s
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  apply proper_sent in Hm; [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: selected_message_exists_in_all_preloaded_traces
  FreeE (field_selector output) s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hplst: valid_state_prop
  (pre_loaded_with_all_messages_vlsm FreeE) s
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m clear Hplst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: selected_message_exists_in_all_preloaded_traces
  FreeE (field_selector output) s m
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  specialize (Hm is tr Htr'pre).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m

  (* Phase II (a): The restriction of tr to the equivocators allowed to
    state-equivocate is valid for the corresponding composition
    pre-loaded with the messages observed in the projection sX of s.
  *)

  specialize
    (finite_valid_trace_sub_projection (equivocator_IM IM) equivocating
      (equivocators_fixed_equivocations_constraint IM Hbs index_listing equivocating)
      (equivocator_Hbs IM Hbs)
      finite_index
      (no_additional_equivocations (free_composite_vlsm IM) sX)
    ) as Hproject.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: ∀ (s : composite_state (equivocator_IM IM)) 
  (tr : list
         (composite_transition_item
         (equivocator_IM IM))),
  trace_sub_item_input_is_seeded_or_sub_previously_sent
    (equivocator_IM IM) equivocating
    (no_additional_equivocations
       (free_composite_vlsm IM) sX) tr
  → finite_valid_trace
      (composite_vlsm 
         (equivocator_IM IM)
         (equivocators_fixed_equivocations_constraint
         IM Hbs index_listing equivocating))
      s tr
    → finite_valid_trace
        (composite_no_equivocation_vlsm_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs))
         (no_additional_equivocations
         (free_composite_vlsm IM) sX))
        (composite_state_sub_projection
         (equivocator_IM IM) equivocating s)
        (finite_trace_sub_projection
         (equivocator_IM IM) equivocating
         tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  spec Hproject is tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) equivocating
  (no_additional_equivocations
     (free_composite_vlsm IM) sX) tr
→ finite_valid_trace
    (composite_vlsm 
       (equivocator_IM IM)
       (equivocators_fixed_equivocations_constraint
         IM Hbs index_listing equivocating))
    is tr
  → finite_valid_trace
      (composite_no_equivocation_vlsm_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs))
         (no_additional_equivocations
         (free_composite_vlsm IM) sX))
      (composite_state_sub_projection
         (equivocator_IM IM) equivocating
         is)
      (finite_trace_sub_projection
         (equivocator_IM IM) equivocating
         tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  spec Hproject.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) equivocating
  (no_additional_equivocations
     (free_composite_vlsm IM) sX) tr
→ finite_valid_trace
    (composite_vlsm 
       (equivocator_IM IM)
       (equivocators_fixed_equivocations_constraint
         IM Hbs index_listing equivocating))
    is tr
  → finite_valid_trace
      (composite_no_equivocation_vlsm_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs))
         (no_additional_equivocations
         (free_composite_vlsm IM) sX))
      (composite_state_sub_projection
         (equivocator_IM IM) equivocating
         is)
      (finite_trace_sub_projection
         (equivocator_IM IM) equivocating
         tr)
trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) equivocating
  (no_additional_equivocations
     (free_composite_vlsm IM) sX) tr
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_vlsm 
     (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint
        IM Hbs index_listing equivocating))
  is tr
→ finite_valid_trace
    (composite_no_equivocation_vlsm_with_pre_loaded
       (sub_IM 
         (equivocator_IM IM) equivocating)
       (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
       (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs))
       (no_additional_equivocations
         (free_composite_vlsm IM) sX))
    (composite_state_sub_projection
       (equivocator_IM IM) equivocating is)
    (finite_trace_sub_projection
       (equivocator_IM IM) equivocating tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) equivocating
  (no_additional_equivocations
     (free_composite_vlsm IM) sX) tr
→ finite_valid_trace
    (composite_vlsm 
       (equivocator_IM IM)
       (equivocators_fixed_equivocations_constraint
         IM Hbs index_listing equivocating))
    is tr
  → finite_valid_trace
      (composite_no_equivocation_vlsm_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs))
         (no_additional_equivocations
         (free_composite_vlsm IM) sX))
      (composite_state_sub_projection
         (equivocator_IM IM) equivocating
         is)
      (finite_trace_sub_projection
         (equivocator_IM IM) equivocating
         tr)
trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) equivocating
  (no_additional_equivocations
     (free_composite_vlsm IM) sX) tr subst sX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) equivocating
  (no_additional_equivocations
     (free_composite_vlsm IM)
     (equivocators_state_project IM
        descriptors s)) tr
→ finite_valid_trace
    (composite_vlsm 
       (equivocator_IM IM)
       (equivocators_fixed_equivocations_constraint
         IM Hbs index_listing equivocating))
    is tr
  → finite_valid_trace
      (composite_no_equivocation_vlsm_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs))
         (no_additional_equivocations
         (free_composite_vlsm IM)
         (equivocators_state_project IM
         descriptors s)))
      (composite_state_sub_projection
         (equivocator_IM IM) equivocating
         is)
      (finite_trace_sub_projection
         (equivocator_IM IM) equivocating
         tr)
trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) equivocating
  (no_additional_equivocations
     (free_composite_vlsm IM)
     (equivocators_state_project IM descriptors s)) tr
    rewrite <- (valid_trace_get_last Htr) in Hdescriptors |- *.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
m: message
Hm: trace_has_message (field_selector output) m tr
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) equivocating
  (no_additional_equivocations
     (free_composite_vlsm IM)
     (equivocators_state_project IM
        descriptors s)) tr
→ finite_valid_trace
    (composite_vlsm 
       (equivocator_IM IM)
       (equivocators_fixed_equivocations_constraint
         IM Hbs index_listing equivocating))
    is tr
  → finite_valid_trace
      (composite_no_equivocation_vlsm_with_pre_loaded
         (sub_IM 
         (equivocator_IM IM) equivocating)
         (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
         (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs))
         (no_additional_equivocations
         (free_composite_vlsm IM)
         (equivocators_state_project IM
         descriptors s)))
      (composite_state_sub_projection
         (equivocator_IM IM) equivocating
         is)
      (finite_trace_sub_projection
         (equivocator_IM IM) equivocating
         tr)
trace_sub_item_input_is_seeded_or_sub_previously_sent
  (equivocator_IM IM) equivocating
  (no_additional_equivocations
     (free_composite_vlsm IM)
     (equivocators_state_project IM descriptors
        (finite_trace_last is tr))) tr
    apply
      (equivocators_trace_sub_item_input_is_seeded_or_sub_previously_sent
        _ _ (valid_trace_forget_last Htr) descriptors Hdescriptors
      ).
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_vlsm 
     (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint
        IM Hbs index_listing equivocating))
  is tr
→ finite_valid_trace
    (composite_no_equivocation_vlsm_with_pre_loaded
       (sub_IM 
         (equivocator_IM IM) equivocating)
       (free_constraint
         (sub_IM 
         (equivocator_IM IM) equivocating))
       (sub_has_been_sent_capabilities
         (equivocator_IM IM) equivocating
         (equivocator_Hbs IM Hbs))
       (no_additional_equivocations
         (free_composite_vlsm IM) sX))
    (composite_state_sub_projection
       (equivocator_IM IM) equivocating is)
    (finite_trace_sub_projection
       (equivocator_IM IM) equivocating tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  spec Hproject (valid_trace_forget_last Htr).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo sX m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m

  rewrite HeqsX in n.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: vstate (free_composite_vlsm IM)
HeqsX: sX =
equivocators_state_project IM descriptors s
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  clear HeqsX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: trace_has_message (field_selector output) m tr
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  (* Phase III (a):
     Obtain a projection trXm of tr outputing m using a final_descriptor_m
  *)

  apply (equivocators_trace_project_output_reflecting_iff _ _ _ (proj1 (valid_trace_forget_last Htr'pre))) in Hm.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
Hm: ∃ (final_descriptors
   initial_descriptors : 
   equivocator_descriptors IM) 
  (trX : list (composite_transition_item IM)),
  not_equivocating_equivocator_descriptors IM
    final_descriptors 
    (finite_trace_last is tr)
  ∧ equivocators_trace_project IM
      final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ Exists (field_selector output m) trX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  destruct Hm as [final_descriptors_m [initial_descriptors_m [trXm [Hfinal_descriptors_m [Hproject_trXm Hex]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
Hex: Exists (field_selector output m) trXm
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m

  (* Identify the item outputing m in trXm an its corresponding item in tr.
  *)

  apply Exists_exists in Hex.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
Hex: ∃ x : transition_item,
  x ∈ trXm ∧ field_selector output m x
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  destruct Hex as [output_itemX [Hin Houtput_select]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
Hin: output_itemX ∈ trXm
Houtput_select: field_selector output m output_itemX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  apply elem_of_list_split in Hin.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
Hin: ∃ l1 l2 : list transition_item,
  trXm = l1 ++ output_itemX :: l2
Houtput_select: field_selector output m output_itemX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  destruct Hin as [preX [sufX Heq_trXm]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ output_itemX :: sufX
Houtput_select: field_selector output m output_itemX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  change (output_itemX :: sufX) with ([output_itemX] ++ sufX) in Heq_trXm.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  assert (Hpr_item := Hproject_trXm).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
Hpr_item: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  rewrite Heq_trXm in Hpr_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
Hpr_item: equivocators_trace_project IM
  final_descriptors_m tr =
Some
  (preX ++ [output_itemX] ++ sufX,
  initial_descriptors_m)
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  apply equivocators_trace_project_app_inv_item in Hpr_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
Hpr_item: ∃ (pre
   suf : list
         (composite_transition_item
         (equivocator_IM IM))) 
  (item : 
   composite_transition_item
     (equivocator_IM IM)) 
  (item_descriptors
   pre_descriptors : 
   equivocator_descriptors IM),
  equivocators_trace_project IM
    final_descriptors_m suf =
  Some (sufX, item_descriptors)
  ∧ equivocators_transition_item_project IM
      item_descriptors item =
    Some
      (Some output_itemX, pre_descriptors)
    ∧ equivocators_trace_project IM
        pre_descriptors pre =
      Some (preX, initial_descriptors_m)
      ∧ tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  destruct Hpr_item as [pre [suf [item [item_descriptors [pre_descriptors [_ [Hpr_item [_ Heqtr]]]]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_item: equivocators_transition_item_project IM
  item_descriptors item =
Some (Some output_itemX, pre_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  unfold equivocators_transition_item_project in Hpr_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
Hpr_item: match
  equivocator_vlsm_transition_item_project
    (IM (projT1 (l item)))
    (composite_transition_item_projection
       (equivocator_IM IM) item)
    (item_descriptors (projT1 (l item)))
with
| Some (Some item', deqv') =>
    Some
      (Some
         {|
         l :=
         existT (projT1 (l item)) (l item');
         input := input item;
         destination :=
         equivocators_state_project IM
         item_descriptors
         (destination item);
         output := output item
         |},
      equivocator_descriptors_update IM
        item_descriptors 
        (projT1 (l item)) deqv')
| Some (None, deqv') =>
    Some
      (None,
      equivocator_descriptors_update IM
        item_descriptors 
        (projT1 (l item)) deqv')
| None => None
end =
Some (Some output_itemX, pre_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  destruct
    (equivocator_vlsm_transition_item_project (IM (projT1 (l item)))
      (composite_transition_item_projection (equivocator_IM IM) item)
      (item_descriptors (projT1 (l item)))
    ) as [(o, deqv')|]
    ; [|congruence].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
o: option (vtransition_item (IM (projT1 (l item))))
deqv': MachineDescriptor (IM (projT1 (l item)))
Hpr_item: match o with
| Some item' =>
    Some
      (Some
         {|
         l :=
         existT (projT1 (l item)) (l item');
         input := input item;
         destination :=
         equivocators_state_project IM
         item_descriptors
         (destination item);
         output := output item
         |},
      equivocator_descriptors_update IM
        item_descriptors 
        (projT1 (l item)) deqv')
| None =>
    Some
      (None,
      equivocator_descriptors_update IM
        item_descriptors 
        (projT1 (l item)) deqv')
end =
Some (Some output_itemX, pre_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  destruct o as [item'|]; [|congruence].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
deqv': MachineDescriptor (IM (projT1 (l item)))
Hpr_item: Some
  (Some
     {|
       l :=
         existT (projT1 (l item)) (l item');
       input := input item;
       destination :=
         equivocators_state_project IM
         item_descriptors
         (destination item);
       output := output item
     |},
  equivocator_descriptors_update IM
    item_descriptors 
    (projT1 (l item)) deqv') =
Some (Some output_itemX, pre_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  inversion Hpr_item.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
output_itemX: transition_item
preX, sufX: list transition_item
Heq_trXm: trXm = preX ++ [output_itemX] ++ sufX
Houtput_select: field_selector output m output_itemX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
item: composite_transition_item (equivocator_IM IM)
item_descriptors, pre_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
deqv': MachineDescriptor (IM (projT1 (l item)))
Hpr_item: Some
  (Some
     {|
       l :=
         existT (projT1 (l item)) (l item');
       input := input item;
       destination :=
         equivocators_state_project IM
         item_descriptors
         (destination item);
       output := output item
     |},
  equivocator_descriptors_update IM
    item_descriptors 
    (projT1 (l item)) deqv') =
Some (Some output_itemX, pre_descriptors)
Heqtr: tr = pre ++ [item] ++ suf
H0: {|
  l := existT (projT1 (l item)) (l item');
  input := input item;
  destination :=
    equivocators_state_project IM
      item_descriptors 
      (destination item);
  output := output item
|} = output_itemX
H1: equivocator_descriptors_update IM
  item_descriptors 
  (projT1 (l item)) deqv' = pre_descriptors
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m subst output_itemX pre_descriptors.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: field_selector output m
  {|
    l :=
      existT 
        (projT1 (l item)) 
        (l item');
    input := input item;
    destination :=
      equivocators_state_project IM
        item_descriptors
        (destination item);
    output := output item
  |}
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
deqv': MachineDescriptor (IM (projT1 (l item)))
Hpr_item: Some
  (Some
     {|
       l :=
         existT (projT1 (l item)) (l item');
       input := input item;
       destination :=
         equivocators_state_project IM
         item_descriptors
         (destination item);
       output := output item
     |},
  equivocator_descriptors_update IM
    item_descriptors 
    (projT1 (l item)) deqv') =
Some
  (Some
     {|
       l :=
         existT (projT1 (l item)) (l item');
       input := input item;
       destination :=
         equivocators_state_project IM
         item_descriptors
         (destination item);
       output := output item
     |},
  equivocator_descriptors_update IM
    item_descriptors 
    (projT1 (l item)) deqv')
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m clear Hpr_item deqv'.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: field_selector output m
  {|
    l :=
      existT 
        (projT1 (l item)) 
        (l item');
    input := input item;
    destination :=
      equivocators_state_project IM
        item_descriptors
        (destination item);
    output := output item
  |}
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  simpl in Houtput_select.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors s
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors s)
    m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m

  (* show that that item must be specifying a transition for an equivocating node*)

  rewrite <- (valid_trace_get_last Htr) in Hdescriptors, n.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors
       (finite_trace_last is tr)) m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  specialize
    (projection_has_not_been_observed_is_equivocating _ _ (valid_trace_forget_last Htr)
      _ Hdescriptors
      _ n item
    ) as Hitem_equivocating.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
Hdescriptors: proper_fixed_equivocator_descriptors
  descriptors 
  (finite_trace_last is tr)
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
n: ¬ composite_has_been_observed IM Hbo
    (equivocators_state_project IM descriptors
       (finite_trace_last is tr)) m
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: item ∈ tr
→ output item = Some m
  → projT1 (l item)
    ∈ equivocating
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  clear Hdescriptors n.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: item ∈ tr
→ output item = Some m
  → projT1 (l item)
    ∈ equivocating
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  spec Hitem_equivocating.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: item ∈ tr
→ output item = Some m
  → projT1 (l item)
    ∈ equivocating
item ∈ tr
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: output item = Some m
→ projT1 (l item) ∈ equivocating
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: item ∈ tr
→ output item = Some m
  → projT1 (l item)
    ∈ equivocating
item ∈ tr subst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m
  (pre ++ [item] ++ suf) =
Some
  (preX ++
   [{|
      l :=
        existT 
         (projT1 (l item)) 
         (l item');
      input := input item;
      destination :=
        equivocators_state_project IM
         item_descriptors
         (destination item);
      output := output item
    |}] ++ sufX,
  initial_descriptors_m)
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating
     (pre ++ [item] ++ suf))
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s (pre ++ [item] ++ suf)
Hitem_equivocating: item ∈ pre ++ [item] ++ suf
→ output item = Some m
  → projT1 (l item)
    ∈ equivocating
item ∈ pre ++ [item] ++ suf
    rewrite !elem_of_app.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m
  (pre ++ [item] ++ suf) =
Some
  (preX ++
   [{|
      l :=
        existT 
         (projT1 (l item)) 
         (l item');
      input := input item;
      destination :=
        equivocators_state_project IM
         item_descriptors
         (destination item);
      output := output item
    |}] ++ sufX,
  initial_descriptors_m)
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating
     (pre ++ [item] ++ suf))
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s (pre ++ [item] ++ suf)
Hitem_equivocating: item ∈ pre ++ [item] ++ suf
→ output item = Some m
  → projT1 (l item)
    ∈ equivocating
item ∈ pre ∨ item ∈ [item] ∨ item ∈ suf
    right.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m
  (pre ++ [item] ++ suf) =
Some
  (preX ++
   [{|
      l :=
        existT 
         (projT1 (l item)) 
         (l item');
      input := input item;
      destination :=
        equivocators_state_project IM
         item_descriptors
         (destination item);
      output := output item
    |}] ++ sufX,
  initial_descriptors_m)
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating
     (pre ++ [item] ++ suf))
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s (pre ++ [item] ++ suf)
Hitem_equivocating: item ∈ pre ++ [item] ++ suf
→ output item = Some m
  → projT1 (l item)
    ∈ equivocating
item ∈ [item] ∨ item ∈ suf left.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m
  (pre ++ [item] ++ suf) =
Some
  (preX ++
   [{|
      l :=
        existT 
         (projT1 (l item)) 
         (l item');
      input := input item;
      destination :=
        equivocators_state_project IM
         item_descriptors
         (destination item);
      output := output item
    |}] ++ sufX,
  initial_descriptors_m)
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating
     (pre ++ [item] ++ suf))
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s (pre ++ [item] ++ suf)
Hitem_equivocating: item ∈ pre ++ [item] ++ suf
→ output item = Some m
  → projT1 (l item)
    ∈ equivocating
item ∈ [item] left. }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: output item = Some m
→ projT1 (l item) ∈ equivocating
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  spec Hitem_equivocating Houtput_select.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m

  (* Phase III (b):
  Consider a projection trX' obtained using the final_descriptor_m as above,
  but first restricting the nodes to just the equivocators allowed to equivocate.
  We will show that we can use [seeded_equivocators_valid_trace_project]
  and leverage the result from Phase II (a)
  to derive that the resulting projection is valid.
  *)
specialize
    (seeded_equivocators_valid_trace_project IM Hbs finite_index equivocating
      (fun m : message =>
        no_additional_equivocations (free_composite_vlsm IM)
          sX m)
      (composite_state_sub_projection (equivocator_IM IM) equivocating is)
      (finite_trace_sub_projection (equivocator_IM IM) equivocating tr)
      Hproject
      (fun i => final_descriptors_m (proj1_sig i))
    ) as Hsub_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        equivocating is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        equivocating tr))
→ ∃ (trX : 
     list
       (composite_transition_item
         (sub_IM IM equivocating))) 
    (initial_descriptors : 
     equivocator_descriptors
       (sub_IM IM equivocating)),
    let isX :=
      equivocators_state_project
        (sub_IM IM equivocating)
        initial_descriptors
        (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is) in
    let final_stateX :=
      finite_trace_last isX trX in
    proper_equivocator_descriptors
      (sub_IM IM equivocating)
      initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
    ∧ equivocators_trace_project
        (sub_IM IM equivocating)
        (λ i : sub_index equivocating,
         final_descriptors_m (`i))
        (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
         (sub_IM IM equivocating)
         (λ i : sub_index equivocating,
         final_descriptors_m (`i))
         (finite_trace_last
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
         (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr)) =
        final_stateX
        ∧ 
        finite_valid_trace
         (pre_loaded_vlsm
         (free_composite_vlsm
         (sub_IM IM equivocating))
         (λ m : message,
         no_additional_equivocations
         (free_composite_vlsm IM) sX m))
         isX trX
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  simpl in Hsub_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        equivocating is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        equivocating tr))
→ ∃ (trX : 
     list
       (composite_transition_item
         (sub_IM IM equivocating))) 
    (initial_descriptors : 
     equivocator_descriptors
       (sub_IM IM equivocating)),
    proper_equivocator_descriptors
      (sub_IM IM equivocating)
      initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
    ∧ equivocators_trace_project
        (sub_IM IM equivocating)
        (λ i : sub_index equivocating,
         final_descriptors_m (`i))
        (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
         (sub_IM IM equivocating)
         (λ i : sub_index equivocating,
         final_descriptors_m (`i))
         (finite_trace_last
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
         (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr)) =
        finite_trace_last
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
        ∧ 
        finite_valid_trace
         (pre_loaded_vlsm
         (free_composite_vlsm
         (sub_IM IM equivocating))
         (λ m : message,
         composite_has_been_observed IM
         Hbo sX m))
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  spec Hsub_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        equivocating is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        equivocating tr))
→ ∃ (trX : 
     list
       (composite_transition_item
         (sub_IM IM equivocating))) 
    (initial_descriptors : 
     equivocator_descriptors
       (sub_IM IM equivocating)),
    proper_equivocator_descriptors
      (sub_IM IM equivocating)
      initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
    ∧ equivocators_trace_project
        (sub_IM IM equivocating)
        (λ i : sub_index equivocating,
         final_descriptors_m (`i))
        (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
         (sub_IM IM equivocating)
         (λ i : sub_index equivocating,
         final_descriptors_m (`i))
         (finite_trace_last
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
         (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr)) =
        finite_trace_last
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
        ∧ 
        finite_valid_trace
         (pre_loaded_vlsm
         (free_composite_vlsm
         (sub_IM IM equivocating))
         (λ m : message,
         composite_has_been_observed IM
         Hbo sX m))
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) equivocating is)
     (finite_trace_sub_projection (equivocator_IM IM)
        equivocating tr))
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: ∃ (trX : list
         (composite_transition_item
         (sub_IM IM equivocating))) 
  (initial_descriptors : 
   equivocator_descriptors
     (sub_IM IM equivocating)),
  proper_equivocator_descriptors
    (sub_IM IM equivocating)
    initial_descriptors
    (composite_state_sub_projection
       (equivocator_IM IM) equivocating
       is)
  ∧ equivocators_trace_project
      (sub_IM IM equivocating)
      (λ i : sub_index equivocating,
         final_descriptors_m (`i))
      (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr) =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project
        (sub_IM IM equivocating)
        (λ i : sub_index equivocating,
         final_descriptors_m (`i))
        (finite_trace_last
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
         (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr)) =
      finite_trace_last
        (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
      ∧ finite_valid_trace
         (pre_loaded_vlsm
         (free_composite_vlsm
         (sub_IM IM equivocating))
         (λ m : message,
         composite_has_been_observed IM
         Hbo sX m))
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        equivocating is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        equivocating tr))
→ ∃ (trX : 
     list
       (composite_transition_item
         (sub_IM IM equivocating))) 
    (initial_descriptors : 
     equivocator_descriptors
       (sub_IM IM equivocating)),
    proper_equivocator_descriptors
      (sub_IM IM equivocating)
      initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
    ∧ equivocators_trace_project
        (sub_IM IM equivocating)
        (λ i : sub_index equivocating,
         final_descriptors_m (`i))
        (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
         (sub_IM IM equivocating)
         (λ i : sub_index equivocating,
         final_descriptors_m (`i))
         (finite_trace_last
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
         (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr)) =
        finite_trace_last
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
        ∧ 
        finite_valid_trace
         (pre_loaded_vlsm
         (free_composite_vlsm
         (sub_IM IM equivocating))
         (λ m : message,
         composite_has_been_observed IM
         Hbo sX m))
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) equivocating is)
     (finite_trace_sub_projection (equivocator_IM IM)
        equivocating tr)) specialize
      (finite_trace_sub_projection_last_state (equivocator_IM IM) equivocating
        (equivocators_fixed_equivocations_constraint IM Hbs index_listing equivocating) _ _ (proj1 (valid_trace_forget_last Htr)))
      as Heq_lst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        equivocating is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        equivocating tr))
→ ∃ (trX : 
     list
       (composite_transition_item
         (sub_IM IM equivocating))) 
    (initial_descriptors : 
     equivocator_descriptors
       (sub_IM IM equivocating)),
    proper_equivocator_descriptors
      (sub_IM IM equivocating)
      initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
    ∧ equivocators_trace_project
        (sub_IM IM equivocating)
        (λ i : sub_index equivocating,
         final_descriptors_m (`i))
        (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
         (sub_IM IM equivocating)
         (λ i : sub_index equivocating,
         final_descriptors_m (`i))
         (finite_trace_last
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
         (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr)) =
        finite_trace_last
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
        ∧ 
        finite_valid_trace
         (pre_loaded_vlsm
         (free_composite_vlsm
         (sub_IM IM equivocating))
         (λ m : message,
         composite_has_been_observed IM
         Hbo sX m))
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
Heq_lst: let lstx := finite_trace_last is tr in
let lstj :=
  finite_trace_last
    (composite_state_sub_projection
       (equivocator_IM IM) equivocating is)
    (finite_trace_sub_projection
       (equivocator_IM IM) equivocating tr)
  in
lstj =
composite_state_sub_projection
  (equivocator_IM IM) equivocating lstx
proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) equivocating is)
     (finite_trace_sub_projection (equivocator_IM IM)
        equivocating tr))
    simpl in Heq_lst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        equivocating is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        equivocating tr))
→ ∃ (trX : 
     list
       (composite_transition_item
         (sub_IM IM equivocating))) 
    (initial_descriptors : 
     equivocator_descriptors
       (sub_IM IM equivocating)),
    proper_equivocator_descriptors
      (sub_IM IM equivocating)
      initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
    ∧ equivocators_trace_project
        (sub_IM IM equivocating)
        (λ i : sub_index equivocating,
         final_descriptors_m (`i))
        (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
         (sub_IM IM equivocating)
         (λ i : sub_index equivocating,
         final_descriptors_m (`i))
         (finite_trace_last
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
         (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr)) =
        finite_trace_last
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
        ∧ 
        finite_valid_trace
         (pre_loaded_vlsm
         (free_composite_vlsm
         (sub_IM IM equivocating))
         (λ m : message,
         composite_has_been_observed IM
         Hbo sX m))
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
Heq_lst: finite_trace_last
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr) =
composite_state_sub_projection
  (equivocator_IM IM) equivocating
  (finite_trace_last is tr)
proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM) equivocating is)
     (finite_trace_sub_projection (equivocator_IM IM)
        equivocating tr))
    match goal with
    |- proper_equivocator_descriptors _ _ ?l =>
      match type of Heq_lst with
      | _ = ?l' =>
        replace l with l'
      end
    end.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        equivocating is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        equivocating tr))
→ ∃ (trX : 
     list
       (composite_transition_item
         (sub_IM IM equivocating))) 
    (initial_descriptors : 
     equivocator_descriptors
       (sub_IM IM equivocating)),
    proper_equivocator_descriptors
      (sub_IM IM equivocating)
      initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
    ∧ equivocators_trace_project
        (sub_IM IM equivocating)
        (λ i : sub_index equivocating,
         final_descriptors_m (`i))
        (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
         (sub_IM IM equivocating)
         (λ i : sub_index equivocating,
         final_descriptors_m (`i))
         (finite_trace_last
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
         (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr)) =
        finite_trace_last
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
        ∧ 
        finite_valid_trace
         (pre_loaded_vlsm
         (free_composite_vlsm
         (sub_IM IM equivocating))
         (λ m : message,
         composite_has_been_observed IM
         Hbo sX m))
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
Heq_lst: finite_trace_last
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr) =
composite_state_sub_projection
  (equivocator_IM IM) equivocating
  (finite_trace_last is tr)
proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (composite_state_sub_projection (equivocator_IM IM)
     equivocating (finite_trace_last is tr))
    intros e.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        equivocating is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        equivocating tr))
→ ∃ (trX : 
     list
       (composite_transition_item
         (sub_IM IM equivocating))) 
    (initial_descriptors : 
     equivocator_descriptors
       (sub_IM IM equivocating)),
    proper_equivocator_descriptors
      (sub_IM IM equivocating)
      initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
    ∧ equivocators_trace_project
        (sub_IM IM equivocating)
        (λ i : sub_index equivocating,
         final_descriptors_m (`i))
        (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
         (sub_IM IM equivocating)
         (λ i : sub_index equivocating,
         final_descriptors_m (`i))
         (finite_trace_last
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
         (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr)) =
        finite_trace_last
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
        ∧ 
        finite_valid_trace
         (pre_loaded_vlsm
         (free_composite_vlsm
         (sub_IM IM equivocating))
         (λ m : message,
         composite_has_been_observed IM
         Hbo sX m))
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
Heq_lst: finite_trace_last
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr) =
composite_state_sub_projection
  (equivocator_IM IM) equivocating
  (finite_trace_last is tr)
e: sub_index equivocating
proper_descriptor (sub_IM IM equivocating e)
  (final_descriptors_m (`e))
  (composite_state_sub_projection (equivocator_IM IM)
     equivocating (finite_trace_last is tr) e)
    destruct e.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        equivocating is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        equivocating tr))
→ ∃ (trX : 
     list
       (composite_transition_item
         (sub_IM IM equivocating))) 
    (initial_descriptors : 
     equivocator_descriptors
       (sub_IM IM equivocating)),
    proper_equivocator_descriptors
      (sub_IM IM equivocating)
      initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
    ∧ equivocators_trace_project
        (sub_IM IM equivocating)
        (λ i : sub_index equivocating,
         final_descriptors_m (`i))
        (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
         (sub_IM IM equivocating)
         (λ i : sub_index equivocating,
         final_descriptors_m (`i))
         (finite_trace_last
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
         (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr)) =
        finite_trace_last
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
        ∧ 
        finite_valid_trace
         (pre_loaded_vlsm
         (free_composite_vlsm
         (sub_IM IM equivocating))
         (λ m : message,
         composite_has_been_observed IM
         Hbo sX m))
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
Heq_lst: finite_trace_last
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr) =
composite_state_sub_projection
  (equivocator_IM IM) equivocating
  (finite_trace_last is tr)
x: index
i: bool_decide (sub_index_prop equivocating x)
proper_descriptor (sub_IM IM equivocating (x ↾ i))
  (final_descriptors_m (`(x ↾ i)))
  (composite_state_sub_projection (equivocator_IM IM)
     equivocating (finite_trace_last is tr) (x ↾ i))
    apply not_equivocating_equivocator_descriptors_proper in Hfinal_descriptors_m.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: proper_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: proper_equivocator_descriptors
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_last
     (composite_state_sub_projection
        (equivocator_IM IM)
        equivocating is)
     (finite_trace_sub_projection
        (equivocator_IM IM)
        equivocating tr))
→ ∃ (trX : 
     list
       (composite_transition_item
         (sub_IM IM equivocating))) 
    (initial_descriptors : 
     equivocator_descriptors
       (sub_IM IM equivocating)),
    proper_equivocator_descriptors
      (sub_IM IM equivocating)
      initial_descriptors
      (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
    ∧ equivocators_trace_project
        (sub_IM IM equivocating)
        (λ i : sub_index equivocating,
         final_descriptors_m (`i))
        (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr) =
      Some (trX, initial_descriptors)
      ∧ equivocators_state_project
         (sub_IM IM equivocating)
         (λ i : sub_index equivocating,
         final_descriptors_m (`i))
         (finite_trace_last
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
         (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr)) =
        finite_trace_last
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
        ∧ 
        finite_valid_trace
         (pre_loaded_vlsm
         (free_composite_vlsm
         (sub_IM IM equivocating))
         (λ m : message,
         composite_has_been_observed IM
         Hbo sX m))
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
Heq_lst: finite_trace_last
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr) =
composite_state_sub_projection
  (equivocator_IM IM) equivocating
  (finite_trace_last is tr)
x: index
i: bool_decide (sub_index_prop equivocating x)
proper_descriptor (sub_IM IM equivocating (x ↾ i))
  (final_descriptors_m (`(x ↾ i)))
  (composite_state_sub_projection (equivocator_IM IM)
     equivocating (finite_trace_last is tr) (x ↾ i))
    apply Hfinal_descriptors_m.
  }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
Hsub_project: ∃ (trX : list
         (composite_transition_item
         (sub_IM IM equivocating))) 
  (initial_descriptors : 
   equivocator_descriptors
     (sub_IM IM equivocating)),
  proper_equivocator_descriptors
    (sub_IM IM equivocating)
    initial_descriptors
    (composite_state_sub_projection
       (equivocator_IM IM) equivocating
       is)
  ∧ equivocators_trace_project
      (sub_IM IM equivocating)
      (λ i : sub_index equivocating,
         final_descriptors_m (`i))
      (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr) =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project
        (sub_IM IM equivocating)
        (λ i : sub_index equivocating,
         final_descriptors_m (`i))
        (finite_trace_last
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)
         (finite_trace_sub_projection
         (equivocator_IM IM)
         equivocating tr)) =
      finite_trace_last
        (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
      ∧ finite_valid_trace
         (pre_loaded_vlsm
         (free_composite_vlsm
         (sub_IM IM equivocating))
         (λ m : message,
         composite_has_been_observed IM
         Hbo sX m))
         (equivocators_state_project
         (sub_IM IM equivocating)
         initial_descriptors
         (composite_state_sub_projection
         (equivocator_IM IM)
         equivocating is)) trX
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  destruct Hsub_project
    as [trX' [initial_descriptors' [_ [Hpr_tr' [_ HtrX]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
Htr'pre: finite_valid_trace_init_to
  (pre_loaded_with_all_messages_vlsm FreeE)
  is s tr
Hproject: finite_valid_trace
  (composite_no_equivocation_vlsm_with_pre_loaded
     (sub_IM 
        (equivocator_IM IM) equivocating)
     (free_constraint
        (sub_IM 
         (equivocator_IM IM) equivocating))
     (sub_has_been_sent_capabilities
        (equivocator_IM IM) equivocating
        (equivocator_Hbs IM Hbs))
     (no_additional_equivocations
        (free_composite_vlsm IM) sX))
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (sub_IM IM equivocating))
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
Hpr_tr': equivocators_trace_project
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr) =
Some (trX', initial_descriptors')
HtrX: finite_valid_trace
  (pre_loaded_vlsm
     (free_composite_vlsm
        (sub_IM IM equivocating))
     (λ m : message,
        composite_has_been_observed IM Hbo sX m))
  (equivocators_state_project
     (sub_IM IM equivocating)
     initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM) equivocating is))
  trX'
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  clear Htr'pre Hproject.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (sub_IM IM equivocating))
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
Hpr_tr': equivocators_trace_project
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr) =
Some (trX', initial_descriptors')
HtrX: finite_valid_trace
  (pre_loaded_vlsm
     (free_composite_vlsm
        (sub_IM IM equivocating))
     (λ m : message,
        composite_has_been_observed IM Hbo sX m))
  (equivocators_state_project
     (sub_IM IM equivocating)
     initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM) equivocating is))
  trX'
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m

  (* State that by restricting trXm to the subset of equivocating nodes
    we obtain the same trX' trace.
  *)

  destruct
    (equivocators_trace_project_finite_trace_sub_projection_commute IM equivocating
      final_descriptors_m initial_descriptors_m initial_descriptors' tr trXm trX'
      Hproject_trXm Hpr_tr'
    )
    as [_ HeqtrX].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
Hproject_trXm: equivocators_trace_project IM
  final_descriptors_m tr =
Some (trXm, initial_descriptors_m)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (sub_IM IM equivocating))
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
Hpr_tr': equivocators_trace_project
  (sub_IM IM equivocating)
  (λ i : sub_index equivocating,
     final_descriptors_m (`i))
  (finite_trace_sub_projection
     (equivocator_IM IM) equivocating tr) =
Some (trX', initial_descriptors')
HtrX: finite_valid_trace
  (pre_loaded_vlsm
     (free_composite_vlsm
        (sub_IM IM equivocating))
     (λ m : message,
        composite_has_been_observed IM Hbo sX m))
  (equivocators_state_project
     (sub_IM IM equivocating)
     initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM) equivocating is))
  trX'
HeqtrX: finite_trace_sub_projection IM equivocating
  trXm = trX'
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  clear Hproject_trXm Hpr_tr'.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (sub_IM IM equivocating))
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
HtrX: finite_valid_trace
  (pre_loaded_vlsm
     (free_composite_vlsm
        (sub_IM IM equivocating))
     (λ m : message,
        composite_has_been_observed IM Hbo sX m))
  (equivocators_state_project
     (sub_IM IM equivocating)
     initial_descriptors'
     (composite_state_sub_projection
        (equivocator_IM IM) equivocating is))
  trX'
HeqtrX: finite_trace_sub_projection IM equivocating
  trXm = trX'
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m

  (* reduce the goal to showing that the message appears in trX'. *)

  remember
    (equivocators_state_project (sub_IM IM equivocating) initial_descriptors'
      (composite_state_sub_projection (equivocator_IM IM) equivocating is)
    ) as isX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (sub_IM IM equivocating))
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
HeqisX: isX =
equivocators_state_project
  (sub_IM IM equivocating)
  initial_descriptors'
  (composite_state_sub_projection
     (equivocator_IM IM) equivocating is)
HtrX: finite_valid_trace
  (pre_loaded_vlsm
     (free_composite_vlsm
        (sub_IM IM equivocating))
     (λ m : message,
        composite_has_been_observed IM Hbo sX m))
  isX trX'
HeqtrX: finite_trace_sub_projection IM equivocating
  trXm = trX'
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m clear HeqisX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (sub_IM IM equivocating))
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
HtrX: finite_valid_trace
  (pre_loaded_vlsm
     (free_composite_vlsm
        (sub_IM IM equivocating))
     (λ m : message,
        composite_has_been_observed IM Hbo sX m))
  isX trX'
HeqtrX: finite_trace_sub_projection IM equivocating
  trXm = trX'
can_emit
  (equivocators_composition_for_observed IM Hbs Hbr
     equivocating sX) m
  apply can_emit_from_valid_trace with isX trX'; [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (sub_IM IM equivocating))
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
HtrX: finite_valid_trace
  (pre_loaded_vlsm
     (free_composite_vlsm
        (sub_IM IM equivocating))
     (λ m : message,
        composite_has_been_observed IM Hbo sX m))
  isX trX'
HeqtrX: finite_trace_sub_projection IM equivocating
  trXm = trX'
trace_has_message (field_selector output) m trX'
  clear HtrX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
tr: list transition_item
Htr: finite_valid_trace_init_to XE is s tr
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
trXm: list (composite_transition_item IM)
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is tr)
preX, sufX: list transition_item
item: composite_transition_item (equivocator_IM IM)
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
Heq_trXm: trXm =
preX ++
[{|
   l := existT (projT1 (l item)) (l item');
   input := input item;
   destination :=
     equivocators_state_project IM
       item_descriptors 
       (destination item);
   output := output item
 |}] ++ sufX
sX: vstate (free_composite_vlsm IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Heqtr: tr = pre ++ [item] ++ suf
Hitem_equivocating: projT1 (l item) ∈ equivocating
trX': list
  (composite_transition_item
     (sub_IM IM equivocating))
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
HeqtrX: finite_trace_sub_projection IM equivocating
  trXm = trX'
trace_has_message (field_selector output) m trX'

  subst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
trace_has_message (field_selector output) m
  (finite_trace_sub_projection IM equivocating
     (preX ++
      [{|
         l := existT (projT1 (l item)) (l item');
         input := input item;
         destination :=
           equivocators_state_project IM
             item_descriptors (destination item);
         output := output item
       |}] ++ sufX))
  rewrite! (finite_trace_sub_projection_app IM).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
trace_has_message (field_selector output) m
  (finite_trace_sub_projection IM equivocating preX ++
   finite_trace_sub_projection IM equivocating
     [{|
        l := existT (projT1 (l item)) (l item');
        input := input item;
        destination :=
          equivocators_state_project IM
            item_descriptors (destination item);
        output := output item
      |}] ++
   finite_trace_sub_projection IM equivocating sufX)
  unfold trace_has_message.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
Exists (field_selector output m)
  (finite_trace_sub_projection IM equivocating preX ++
   finite_trace_sub_projection IM equivocating
     [{|
        l := existT (projT1 (l item)) (l item');
        input := input item;
        destination :=
          equivocators_state_project IM
            item_descriptors (destination item);
        output := output item
      |}] ++
   finite_trace_sub_projection IM equivocating sufX)
  rewrite! Exists_app.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
Exists (field_selector output m)
  (finite_trace_sub_projection IM equivocating preX)
∨ Exists (field_selector output m)
    (finite_trace_sub_projection IM equivocating
       [{|
          l := existT (projT1 (l item)) (l item');
          input := input item;
          destination :=
            equivocators_state_project IM
              item_descriptors (destination item);
          output := output item
        |}])
  ∨ Exists (field_selector output m)
      (finite_trace_sub_projection IM equivocating
         sufX) right.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
Exists (field_selector output m)
  (finite_trace_sub_projection IM equivocating
     [{|
        l := existT (projT1 (l item)) (l item');
        input := input item;
        destination :=
          equivocators_state_project IM
            item_descriptors (destination item);
        output := output item
      |}])
∨ Exists (field_selector output m)
    (finite_trace_sub_projection IM equivocating sufX) left.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
Exists (field_selector output m)
  (finite_trace_sub_projection IM equivocating
     [{|
        l := existT (projT1 (l item)) (l item');
        input := input item;
        destination :=
          equivocators_state_project IM
            item_descriptors (destination item);
        output := output item
      |}]) simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
Exists (field_selector output m)
  match
    pre_VLSM_projection_transition_item_project
      (composite_type IM)
      (composite_type (sub_IM IM equivocating))
      (composite_label_sub_projection_option IM
         equivocating)
      (composite_state_sub_projection IM equivocating)
      {|
        l := existT (projT1 (l item)) (l item');
        input := input item;
        destination :=
          equivocators_state_project IM
            item_descriptors (destination item);
        output := output item
      |}
  with
  | Some b => [b]
  | None => []
  end

  unfold pre_VLSM_projection_transition_item_project,
    composite_label_sub_projection_option.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
Exists (field_selector output m)
  match
    match
      match
        decide
          (projT1
             (l
                {|
                  l :=
                    existT (projT1 (l item)) (l item');
                  input := input item;
                  destination :=
                    equivocators_state_project IM
                      item_descriptors
                      (destination item);
                  output := output item
                |}) ∈ equivocating)
      with
      | left i_in =>
          Some
            (composite_label_sub_projection IM
               equivocating
               (l
                  {|
                    l :=
                      existT (projT1 (l item))
                        (l item');
                    input := input item;
                    destination :=
                      equivocators_state_project IM
                        item_descriptors
                        (destination item);
                    output := output item
                  |}) i_in)
      | right _ => None
      end
    with
    | Some lY =>
        Some
          {|
            l := lY;
            input :=
              input
                {|
                  l :=
                    existT (projT1 (l item)) (l item');
                  input := input item;
                  destination :=
                    equivocators_state_project IM
                      item_descriptors
                      (destination item);
                  output := output item
                |};
            destination :=
              composite_state_sub_projection IM
                equivocating
                (destination
                   {|
                     l :=
                       existT (projT1 (l item))
                         (l item');
                     input := input item;
                     destination :=
                       equivocators_state_project IM
                         item_descriptors
                         (destination item);
                     output := output item
                   |});
            output :=
              output
                {|
                  l :=
                    existT (projT1 (l item)) (l item');
                  input := input item;
                  destination :=
                    equivocators_state_project IM
                      item_descriptors
                      (destination item);
                  output := output item
                |}
          |}
    | None => None
    end
  with
  | Some b => [b]
  | None => []
  end
  simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
Exists (field_selector output m)
  match
    match
      match
        decide (projT1 (l item) ∈ equivocating)
      with
      | left i_in =>
          Some
            (composite_label_sub_projection IM
               equivocating
               (existT (projT1 (l item)) (l item'))
               i_in)
      | right _ => None
      end
    with
    | Some lY =>
        Some
          {|
            l := lY;
            input := input item;
            destination :=
              composite_state_sub_projection IM
                equivocating
                (equivocators_state_project IM
                   item_descriptors (destination item));
            output := output item
          |}
    | None => None
    end
  with
  | Some b => [b]
  | None => []
  end
  case_decide; [|contradiction].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
H: projT1 (l item) ∈ equivocating
Exists (field_selector output m)
  [{|
     l :=
       composite_label_sub_projection IM equivocating
         (existT (projT1 (l item)) (l item')) H;
     input := input item;
     destination :=
       composite_state_sub_projection IM equivocating
         (equivocators_state_project IM
            item_descriptors (destination item));
     output := output item
   |}]
  constructor.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
H: projT1 (l item) ∈ equivocating
field_selector output m
  {|
    l :=
      composite_label_sub_projection IM equivocating
        (existT (projT1 (l item)) (l item')) H;
    input := input item;
    destination :=
      composite_state_sub_projection IM equivocating
        (equivocators_state_project IM
           item_descriptors (destination item));
    output := output item
  |} simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
s: composite_state (equivocator_IM IM)
is: state
item: composite_transition_item (equivocator_IM IM)
pre, suf: list
  (composite_transition_item (equivocator_IM IM))
Htr: finite_valid_trace_init_to XE is s
  (pre ++ [item] ++ suf)
descriptors: equivocator_descriptors IM
m: message
final_descriptors_m, initial_descriptors_m: equivocator_descriptors IM
Hfinal_descriptors_m: not_equivocating_equivocator_descriptors
  IM final_descriptors_m
  (finite_trace_last is
     (pre ++ [item] ++ suf))
preX, sufX: list transition_item
item_descriptors: equivocator_descriptors IM
item': vtransition_item (IM (projT1 (l item)))
Houtput_select: output item = Some m
sX: vstate (free_composite_vlsm IM)
Hitem_equivocating: projT1 (l item) ∈ equivocating
initial_descriptors': equivocator_descriptors
  (sub_IM IM equivocating)
isX: vstate
  (free_composite_vlsm (sub_IM IM equivocating))
H: projT1 (l item) ∈ equivocating
output item = Some m assumption.
Qed.

Main result of this section, stating that traces which are valid for the equivocator-based definition of fixed equivocation project to traces which are valid for the simple-nodes definition of fixed equivocation.

Theorem fixed_equivocators_valid_trace_project
  (final_descriptors : equivocator_descriptors IM)
  (is : composite_state (equivocator_IM IM))
  (tr : list (composite_transition_item (equivocator_IM IM)))
  (final_state := finite_trace_last is tr)
  (Hproper: proper_fixed_equivocator_descriptors final_descriptors final_state)
  (Htr : finite_valid_trace XE is tr)
  : exists
    (trX : list (composite_transition_item IM))
    (initial_descriptors : equivocator_descriptors IM)
    (isX := equivocators_state_project IM initial_descriptors is)
    (final_stateX := finite_trace_last isX trX),
    proper_fixed_equivocator_descriptors initial_descriptors is /\
    equivocators_trace_project IM final_descriptors tr = Some (trX, initial_descriptors) /\
    equivocators_state_project IM final_descriptors final_state = final_stateX /\
    finite_valid_trace X isX trX.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X isX trX
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
∃ (trX : list (composite_transition_item IM)) (initial_descriptors : 
                                               equivocator_descriptors
                                                 IM),
  let isX :=
    equivocators_state_project IM initial_descriptors
      is in
  let final_stateX := finite_trace_last isX trX in
  proper_fixed_equivocator_descriptors
    initial_descriptors is
  ∧ equivocators_trace_project IM final_descriptors tr =
    Some (trX, initial_descriptors)
    ∧ equivocators_state_project IM final_descriptors
        final_state = final_stateX
      ∧ finite_valid_trace X isX trX
  apply _equivocators_valid_trace_project; [assumption | assumption| ..]
  ; intros.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
l: composite_label IM
s: composite_state IM
fixed_equivocation_constraint IM Hbs Hbr equivocating
  l (s, None)
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
constraint_has_been_sent_prop
  (fixed_equivocation_constraint IM Hbs Hbr
     equivocating)
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
l: composite_label IM
s: composite_state IM
fixed_equivocation_constraint IM Hbs Hbr equivocating
  l (s, None) exact I.
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
is: composite_state (equivocator_IM IM)
tr: list
  (composite_transition_item (equivocator_IM IM))
final_state:= finite_trace_last is tr: state
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors final_state
Htr: finite_valid_trace XE is tr
constraint_has_been_sent_prop
  (fixed_equivocation_constraint IM Hbs Hbr
     equivocating) apply fixed_equivocation_constraint_has_constraint_has_been_sent_prop.
Qed.

Lemma fixed_equivocators_vlsm_partial_projection
  (final_descriptors : equivocator_descriptors IM)
  : VLSM_partial_projection XE X (equivocators_partial_trace_project IM finite_index final_descriptors).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
VLSM_partial_projection XE X
  (equivocators_partial_trace_project IM finite_index
     final_descriptors)
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
VLSM_partial_projection XE X
  (equivocators_partial_trace_project IM finite_index
     final_descriptors)
  split; [split|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
∀ (sX : vstate XE) (trX : list (vtransition_item XE)) 
  (sY : vstate X) (trY : list (vtransition_item X)),
  equivocators_partial_trace_project IM finite_index
    final_descriptors (sX, trX) = Some (sY, trY)
  → ∀ (s'X : state) (preX : list transition_item),
      finite_trace_last s'X preX = sX
      → finite_valid_trace_from XE s'X (preX ++ trX)
        → ∃ (s'Y : vstate X) (preY : list
                                       (vtransition_item
                                          X)),
            equivocators_partial_trace_project IM
              finite_index final_descriptors
              (s'X, preX ++ trX) =
            Some (s'Y, preY ++ trY)
            ∧ finite_trace_last s'Y preY = sY
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
∀ (sX : vstate XE) 
  (trX : list (vtransition_item XE)) 
  (sY : vstate X) 
  (trY : list (vtransition_item X)),
  equivocators_partial_trace_project IM finite_index
    final_descriptors 
    (sX, trX) = 
  Some (sY, trY)
  → finite_valid_trace XE sX trX
    → finite_valid_trace X sY trY
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
∀ (sX : vstate XE) (trX : list (vtransition_item XE)) 
  (sY : vstate X) (trY : list (vtransition_item X)),
  equivocators_partial_trace_project IM finite_index
    final_descriptors (sX, trX) = Some (sY, trY)
  → ∀ (s'X : state) (preX : list transition_item),
      finite_trace_last s'X preX = sX
      → finite_valid_trace_from XE s'X (preX ++ trX)
        → ∃ (s'Y : vstate X) (preY : list
                                       (vtransition_item
                                          X)),
            equivocators_partial_trace_project IM
              finite_index final_descriptors
              (s'X, preX ++ trX) =
            Some (s'Y, preY ++ trY)
            ∧ finite_trace_last s'Y preY = sY intros s tr sX trX Hpr_tr s_pre pre Hs_lst Hpre_tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
s_pre: state
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from XE s_pre (pre ++ tr)
∃ (s'Y : vstate X) (preY : list (vtransition_item X)),
  equivocators_partial_trace_project IM finite_index
    final_descriptors (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
    assert
      (HPreFree_pre_tr : finite_valid_trace_from (pre_loaded_with_all_messages_vlsm FreeE) s_pre (pre ++ tr)).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
s_pre: state
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from XE s_pre (pre ++ tr)
finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) s_pre
  (pre ++ tr)
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
s_pre: state
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from XE s_pre (pre ++ tr)
HPreFree_pre_tr: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) s_pre 
  (pre ++ tr)
∃ (s'Y : vstate X) (preY : list (vtransition_item X)),
  equivocators_partial_trace_project IM finite_index
    final_descriptors 
    (s_pre, pre ++ tr) = 
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
    {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
s_pre: state
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from XE s_pre (pre ++ tr)
finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm FreeE) s_pre
  (pre ++ tr) revert Hpre_tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
s_pre: state
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
finite_valid_trace_from XE s_pre (pre ++ tr)
→ finite_valid_trace_from
    (pre_loaded_with_all_messages_vlsm FreeE) s_pre
    (pre ++ tr) apply VLSM_incl_finite_valid_trace_from.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
s_pre: state
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
VLSM_incl_part
  (composite_vlsm_machine (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint IM
        Hbs index_listing equivocating))
  (pre_loaded_with_all_messages_vlsm_machine FreeE)
      apply equivocators_fixed_equivocations_vlsm_incl_PreFree.
    }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
s_pre: state
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
Hpre_tr: finite_valid_trace_from XE s_pre (pre ++ tr)
HPreFree_pre_tr: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) s_pre 
  (pre ++ tr)
∃ (s'Y : vstate X) (preY : list (vtransition_item X)),
  equivocators_partial_trace_project IM finite_index
    final_descriptors (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX
    clear Hpre_tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
s_pre: state
pre: list transition_item
Hs_lst: finite_trace_last s_pre pre = s
HPreFree_pre_tr: finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     FreeE) s_pre 
  (pre ++ tr)
∃ (s'Y : vstate X) (preY : list (vtransition_item X)),
  equivocators_partial_trace_project IM finite_index
    final_descriptors (s_pre, pre ++ tr) =
  Some (s'Y, preY ++ trX)
  ∧ finite_trace_last s'Y preY = sX  revert s tr sX trX Hpr_tr s_pre pre Hs_lst HPreFree_pre_tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
∀ (s : vstate XE) (tr : list (vtransition_item XE)) 
  (sX : vstate X) (trX : list (vtransition_item X)),
  equivocators_partial_trace_project IM finite_index
    final_descriptors (s, tr) = Some (sX, trX)
  → ∀ (s_pre : state) (pre : list transition_item),
      finite_trace_last s_pre pre = s
      → finite_valid_trace_from
          (pre_loaded_with_all_messages_vlsm FreeE)
          s_pre (pre ++ tr)
        → ∃ (s'Y : vstate X) (preY : list
                                       (vtransition_item
                                          X)),
            equivocators_partial_trace_project IM
              finite_index final_descriptors
              (s_pre, pre ++ tr) =
            Some (s'Y, preY ++ trX)
            ∧ finite_trace_last s'Y preY = sX
    apply equivocators_partial_trace_project_extends_left.
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
∀ (sX : vstate XE) (trX : list (vtransition_item XE)) 
  (sY : vstate X) (trY : list (vtransition_item X)),
  equivocators_partial_trace_project IM finite_index
    final_descriptors (sX, trX) = Some (sY, trY)
  → finite_valid_trace XE sX trX
    → finite_valid_trace X sY trY intros s tr sX trX Hpr_tr Htr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace XE s tr
finite_valid_trace X sX trX
    destruct (destruct_equivocators_partial_trace_project IM finite_index Hpr_tr)
      as [Hnot_equiv [initial_descriptors [Htr_project Hs_project]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace XE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
finite_valid_trace X sX trX

    specialize (finite_valid_trace_last_pstate XE _ _ (proj1 Htr)) as Hlst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace XE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hlst: valid_state_prop XE (finite_trace_last s tr)
finite_valid_trace X sX trX
    apply not_equivocating_equivocator_descriptors_proper_fixed in Hnot_equiv as Hproper
    ; [|assumption].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace XE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hlst: valid_state_prop XE (finite_trace_last s tr)
finite_valid_trace X sX trX
    destruct (fixed_equivocators_valid_trace_project  _ _ _ Hproper Htr)
      as [_trX [_initial_descriptors [_ [_Htr_project [_ HtrX]]]]].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace XE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hlst: valid_state_prop XE (finite_trace_last s tr)
_trX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace X
  (equivocators_state_project IM
     _initial_descriptors s) _trX
finite_valid_trace X sX trX
    rewrite Htr_project in _Htr_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace XE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hlst: valid_state_prop XE (finite_trace_last s tr)
_trX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Htr_project: Some (trX, initial_descriptors) =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace X
  (equivocators_state_project IM
     _initial_descriptors s) _trX
finite_valid_trace X sX trX
    inversion _Htr_project.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
sX: vstate X
trX: list (vtransition_item X)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) = 
Some (sX, trX)
Htr: finite_valid_trace XE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (finite_trace_last s tr)
initial_descriptors: equivocator_descriptors IM
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (trX, initial_descriptors)
Hs_project: equivocators_state_project IM
  initial_descriptors s = sX
Hlst: valid_state_prop XE (finite_trace_last s tr)
_trX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Htr_project: Some (trX, initial_descriptors) =
Some (_trX, _initial_descriptors)
HtrX: finite_valid_trace X
  (equivocators_state_project IM
     _initial_descriptors s) _trX
H0: trX = _trX
H1: initial_descriptors = _initial_descriptors
finite_valid_trace X sX _trX subst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
final_descriptors: equivocator_descriptors IM
s: vstate XE
tr: list (vtransition_item XE)
_trX: list (composite_transition_item IM)
_initial_descriptors: equivocator_descriptors IM
_Htr_project: Some (_trX, _initial_descriptors) =
Some (_trX, _initial_descriptors)
Hpr_tr: equivocators_partial_trace_project IM
  finite_index final_descriptors 
  (s, tr) =
Some
  (equivocators_state_project IM
     _initial_descriptors s, _trX)
Htr: finite_valid_trace XE s tr
Hnot_equiv: not_equivocating_equivocator_descriptors
  IM final_descriptors
  (finite_trace_last s tr)
Hproper: proper_fixed_equivocator_descriptors
  final_descriptors 
  (finite_trace_last s tr)
Htr_project: equivocators_trace_project IM
  final_descriptors tr =
Some (_trX, _initial_descriptors)
Hlst: valid_state_prop XE (finite_trace_last s tr)
HtrX: finite_valid_trace X
  (equivocators_state_project IM
     _initial_descriptors s) _trX
finite_valid_trace X
  (equivocators_state_project IM _initial_descriptors
     s) _trX  assumption.
Qed.

Lemma fixed_equivocators_vlsm_projection
  : VLSM_projection XE X (equivocators_total_label_project IM) (equivocators_total_state_project IM).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
VLSM_projection XE X
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM)
Proof.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
VLSM_projection XE X
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM)
  constructor; [constructor|]; intros.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace_from XE sX trX
equivocators_total_state_project IM
  (finite_trace_last sX trX) =
finite_trace_last
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_trace_project (type XE)
     (type X) (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_trace_project 
     (type XE) 
     (type X) 
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace_from XE sX trX
equivocators_total_state_project IM
  (finite_trace_last sX trX) =
finite_trace_last
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_trace_project (type XE)
     (type X) (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX) apply PreFreeE_Free_vlsm_projection_type.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace_from XE sX trX
finite_valid_trace_from
  (pre_loaded_with_all_messages_vlsm
     (free_composite_vlsm (equivocator_IM IM))) sX trX
    revert H.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
finite_valid_trace_from XE sX trX
→ finite_valid_trace_from
    (pre_loaded_with_all_messages_vlsm
       (free_composite_vlsm (equivocator_IM IM))) sX
    trX apply VLSM_incl_finite_valid_trace_from.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
VLSM_incl_part
  (composite_vlsm_machine (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint IM
        Hbs index_listing equivocating))
  (pre_loaded_with_all_messages_vlsm_machine
     (free_composite_vlsm (equivocator_IM IM)))
    apply equivocators_fixed_equivocations_vlsm_incl_PreFree.
  -message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_trace_project (type XE)
     (type X) (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX) assert (Hpre_tr : finite_valid_trace (pre_loaded_with_all_messages_vlsm FreeE) sX trX).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE) sX trX
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_trace_project 
     (type XE) 
     (type X) 
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE) sX trX revert H.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
finite_valid_trace XE sX trX
→ finite_valid_trace
    (pre_loaded_with_all_messages_vlsm FreeE) sX trX apply VLSM_incl_finite_valid_trace.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
VLSM_incl_part
  (composite_vlsm_machine (equivocator_IM IM)
     (equivocators_fixed_equivocations_constraint IM
        Hbs index_listing equivocating))
  (pre_loaded_with_all_messages_vlsm_machine FreeE)
      apply equivocators_fixed_equivocations_vlsm_incl_PreFree.
    }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_trace_project (type XE)
     (type X) (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    specialize
     (VLSM_partial_projection_finite_valid_trace (fixed_equivocators_vlsm_partial_projection (zero_descriptor IM))
       sX trX (equivocators_state_project IM (zero_descriptor IM) sX) (equivocators_total_trace_project IM trX)
     ) as Hsim.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
Hsim: equivocators_partial_trace_project IM
  finite_index 
  (zero_descriptor IM) 
  (sX, trX) =
Some
  (equivocators_state_project IM
     (zero_descriptor IM) sX,
  equivocators_total_trace_project IM trX)
→ finite_valid_trace XE sX trX
  → finite_valid_trace X
      (equivocators_state_project IM
         (zero_descriptor IM) sX)
      (equivocators_total_trace_project IM trX)
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_trace_project (type XE)
     (type X) (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    spec Hsim.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
Hsim: equivocators_partial_trace_project IM
  finite_index 
  (zero_descriptor IM) 
  (sX, trX) =
Some
  (equivocators_state_project IM
     (zero_descriptor IM) sX,
  equivocators_total_trace_project IM trX)
→ finite_valid_trace XE sX trX
  → finite_valid_trace X
      (equivocators_state_project IM
         (zero_descriptor IM) sX)
      (equivocators_total_trace_project IM trX)
equivocators_partial_trace_project IM finite_index
  (zero_descriptor IM) (sX, trX) =
Some
  (equivocators_state_project IM (zero_descriptor IM)
     sX, equivocators_total_trace_project IM trX)
message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability
  Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability
  Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_trace_project 
     (type XE) 
     (type X) 
     (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    {message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
Hsim: equivocators_partial_trace_project IM
  finite_index 
  (zero_descriptor IM) 
  (sX, trX) =
Some
  (equivocators_state_project IM
     (zero_descriptor IM) sX,
  equivocators_total_trace_project IM trX)
→ finite_valid_trace XE sX trX
  → finite_valid_trace X
      (equivocators_state_project IM
         (zero_descriptor IM) sX)
      (equivocators_total_trace_project IM trX)
equivocators_partial_trace_project IM finite_index
  (zero_descriptor IM) (sX, trX) =
Some
  (equivocators_state_project IM (zero_descriptor IM)
     sX, equivocators_total_trace_project IM trX) simpl.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
Hsim: equivocators_partial_trace_project IM
  finite_index 
  (zero_descriptor IM) 
  (sX, trX) =
Some
  (equivocators_state_project IM
     (zero_descriptor IM) sX,
  equivocators_total_trace_project IM trX)
→ finite_valid_trace XE sX trX
  → finite_valid_trace X
      (equivocators_state_project IM
         (zero_descriptor IM) sX)
      (equivocators_total_trace_project IM trX)
(if
  decide
    (not_equivocating_equivocator_descriptors IM
       (zero_descriptor IM) (finite_trace_last sX trX))
 then
  match
    equivocators_trace_project IM (zero_descriptor IM)
      trX
  with
  | Some (trX, initial_descriptors) =>
      Some
        (equivocators_state_project IM
           initial_descriptors sX, trX)
  | None => None
  end
 else None) =
Some
  (equivocators_state_project IM (zero_descriptor IM)
     sX, equivocators_total_trace_project IM trX) rewrite decide_True by apply zero_descriptor_not_equivocating.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
Hsim: equivocators_partial_trace_project IM
  finite_index 
  (zero_descriptor IM) 
  (sX, trX) =
Some
  (equivocators_state_project IM
     (zero_descriptor IM) sX,
  equivocators_total_trace_project IM trX)
→ finite_valid_trace XE sX trX
  → finite_valid_trace X
      (equivocators_state_project IM
         (zero_descriptor IM) sX)
      (equivocators_total_trace_project IM trX)
match
  equivocators_trace_project IM (zero_descriptor IM)
    trX
with
| Some (trX, initial_descriptors) =>
    Some
      (equivocators_state_project IM
         initial_descriptors sX, trX)
| None => None
end =
Some
  (equivocators_state_project IM (zero_descriptor IM)
     sX, equivocators_total_trace_project IM trX)
      rewrite (equivocators_total_trace_project_characterization IM (proj1 Hpre_tr)).message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
Hsim: equivocators_partial_trace_project IM
  finite_index 
  (zero_descriptor IM) 
  (sX, trX) =
Some
  (equivocators_state_project IM
     (zero_descriptor IM) sX,
  equivocators_total_trace_project IM trX)
→ finite_valid_trace XE sX trX
  → finite_valid_trace X
      (equivocators_state_project IM
         (zero_descriptor IM) sX)
      (equivocators_total_trace_project IM trX)
Some
  (equivocators_state_project IM (zero_descriptor IM)
     sX, equivocators_total_trace_project IM trX) =
Some
  (equivocators_state_project IM (zero_descriptor IM)
     sX, equivocators_total_trace_project IM trX)
      reflexivity.
    }message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace XE sX trX
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_trace_project (type XE)
     (type X) (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    apply Hsim in H.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace X
  (equivocators_state_project IM
     (zero_descriptor IM) sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
finite_valid_trace X
  (equivocators_total_state_project IM sX)
  (pre_VLSM_projection_trace_project (type XE)
     (type X) (equivocators_total_label_project IM)
     (equivocators_total_state_project IM) trX)
    remember (pre_VLSM_projection_trace_project _ _ _ _ _) as tr.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace X
  (equivocators_state_project IM
     (zero_descriptor IM) sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
tr: list transition_item
Heqtr: tr =
pre_VLSM_projection_trace_project 
  (type XE) 
  (type X)
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX
finite_valid_trace X
  (equivocators_total_state_project IM sX) tr
    replace tr with (equivocators_total_trace_project IM trX); [assumption|].message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace X
  (equivocators_state_project IM
     (zero_descriptor IM) sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
tr: list transition_item
Heqtr: tr =
pre_VLSM_projection_trace_project 
  (type XE) 
  (type X)
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX
equivocators_total_trace_project IM trX = tr
    subst.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace X
  (equivocators_state_project IM
     (zero_descriptor IM) sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
equivocators_total_trace_project IM trX =
pre_VLSM_projection_trace_project (type XE) (type X)
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX symmetry.message: Type
EqDecision0: EqDecision message
index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
Hbr: ∀ i : index, HasBeenReceivedCapability (IM i)
Hbo:= λ i : index,
  HasBeenObservedCapability_from_sent_received
    (IM i): ∀ i : index, HasBeenObservedCapability (IM i)
equivocating: set index
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing equivocating: VLSM message
X:= fixed_equivocation_vlsm_composition IM Hbs Hbr
  equivocating: VLSM message
equivocators_free_Hbs:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  (free_composite_vlsm
     (equivocator_IM IM))
FreeE:= free_composite_vlsm (equivocator_IM IM): VLSM message
FreeE_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  (equivocator_IM IM)
  finite_index
  (equivocator_Hbs IM Hbs): HasBeenSentCapability
  FreeE
Hdec_init: ∀ i : index,
  vdecidable_initial_messages_prop (IM i)
Free:= free_composite_vlsm IM: VLSM message
Free_HasBeenSentCapability:= free_composite_HasBeenSentCapability
  IM finite_index Hbs: HasBeenSentCapability
  Free
Free_HasBeenReceivedCapability:= free_composite_HasBeenReceivedCapability
  IM finite_index Hbr: HasBeenReceivedCapability Free
Free_HasBeenObservedCapability:= free_composite_HasBeenObservedCapability
  IM finite_index Hbo: HasBeenObservedCapability Free
Free_no_additional_equivocation_decidable:= no_additional_equivocations_dec
  Free: RelDecision
  (no_additional_equivocations
     Free)
index_equivocating_prop:= sub_index_prop equivocating: index → Prop
equivocating_index:= sub_index equivocating: Type
equivocating_IM:= sub_IM IM equivocating: sub_index equivocating
→ VLSM message
free_equivocating_vlsm_composition:= free_composite_vlsm
  equivocating_IM: VLSM message
sub_equivocator_IM:= sub_IM (equivocator_IM IM)
  equivocating: sub_index equivocating
→ VLSM message
sX: state
trX: list transition_item
H: finite_valid_trace X
  (equivocators_state_project IM
     (zero_descriptor IM) sX)
  (equivocators_total_trace_project IM trX)
Hpre_tr: finite_valid_trace
  (pre_loaded_with_all_messages_vlsm FreeE)
  sX trX
Hsim: finite_valid_trace XE sX trX
→ finite_valid_trace X
    (equivocators_state_project IM
       (zero_descriptor IM) sX)
    (equivocators_total_trace_project IM trX)
pre_VLSM_projection_trace_project (type XE) (type X)
  (equivocators_total_label_project IM)
  (equivocators_total_state_project IM) trX =
equivocators_total_trace_project IM trX
    apply (equivocators_total_VLSM_projection_trace_project IM (proj1 Hpre_tr)).
Qed.

End from_equivocators_to_nodes.

Section all_equivocating.

Fixed Equivocation for all Equivocators

In this section we show that if the set of nodes allowed to equivocate in the fixed set equivocation is the entire set of nodes, then the composition under the fixed-set equivocation constraint is the same as the composition with only the no-message-equivocation constraint.

Context
  {message : Type}
  {index : Type}
  {IndEqDec : EqDecision index}
  (IM : index -> VLSM message)
  (Hbs : forall i : index, HasBeenSentCapability (IM i))
  {index_listing : list index}
  (finite_index : Listing index_listing)
  (XE : VLSM message := equivocators_fixed_equivocations_vlsm IM Hbs index_listing index_listing)
  (NE : VLSM message := equivocators_no_equivocations_vlsm IM Hbs)
  .

Lemma strong_constraint_subsumption_fixed_all
  : strong_constraint_subsumption (equivocator_IM IM)
    (equivocators_no_equivocations_constraint IM Hbs)
    (equivocators_fixed_equivocations_constraint IM Hbs index_listing index_listing).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
strong_constraint_subsumption (equivocator_IM IM)
  (equivocators_no_equivocations_constraint IM Hbs)
  (equivocators_fixed_equivocations_constraint IM Hbs
     index_listing index_listing)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
strong_constraint_subsumption (equivocator_IM IM)
  (equivocators_no_equivocations_constraint IM Hbs)
  (equivocators_fixed_equivocations_constraint IM Hbs
     index_listing index_listing)
  split; [assumption|].message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
l: composite_label (equivocator_IM IM)
som: (composite_state (equivocator_IM IM) *
 option message)%type
H: equivocators_no_equivocations_constraint IM Hbs l
  som
state_has_fixed_equivocation IM index_listing
  index_listing
  (composite_transition (equivocator_IM IM) l som).1
  intro; intros.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
l: composite_label (equivocator_IM IM)
som: (composite_state (equivocator_IM IM) *
 option message)%type
H: equivocators_no_equivocations_constraint IM Hbs l
  som
x: index
H0: x
∈ equivocating_indices IM index_listing
    (composite_transition 
       (equivocator_IM IM) l som).1
x ∈ index_listing apply elem_of_list_In.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
l: composite_label (equivocator_IM IM)
som: (composite_state (equivocator_IM IM) *
 option message)%type
H: equivocators_no_equivocations_constraint IM Hbs l
  som
x: index
H0: x
∈ equivocating_indices IM index_listing
    (composite_transition 
       (equivocator_IM IM) l som).1
In x index_listing apply finite_index.
Qed.

Lemma strong_constraint_subsumption_all_fixed
  : strong_constraint_subsumption (equivocator_IM IM)
    (equivocators_fixed_equivocations_constraint IM Hbs index_listing index_listing)
    (equivocators_no_equivocations_constraint IM Hbs).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
strong_constraint_subsumption (equivocator_IM IM)
  (equivocators_fixed_equivocations_constraint IM Hbs
     index_listing index_listing)
  (equivocators_no_equivocations_constraint IM Hbs)
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
strong_constraint_subsumption (equivocator_IM IM)
  (equivocators_fixed_equivocations_constraint IM Hbs
     index_listing index_listing)
  (equivocators_no_equivocations_constraint IM Hbs)
  intros l (s, om) Hv.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
l: composite_label (equivocator_IM IM)
s: composite_state (equivocator_IM IM)
om: option message
Hv: equivocators_fixed_equivocations_constraint IM
  Hbs index_listing index_listing l 
  (s, om)
equivocators_no_equivocations_constraint IM Hbs l
  (s, om) apply Hv.
Qed.

Lemma equivocators_fixed_equivocations_all_eq : VLSM_eq XE NE.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
VLSM_eq XE NE
Proof.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
VLSM_eq XE NE
  apply VLSM_eq_incl_iff.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
VLSM_incl
  (mk_vlsm
     (composite_vlsm_machine (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM Hbs index_listing index_listing)))
  (mk_vlsm
     (composite_vlsm_machine (equivocator_IM IM)
        (equivocators_no_equivocations_constraint IM
           Hbs)))
∧ VLSM_incl
    (mk_vlsm
       (composite_vlsm_machine (equivocator_IM IM)
          (equivocators_no_equivocations_constraint IM
             Hbs)))
    (mk_vlsm
       (composite_vlsm_machine (equivocator_IM IM)
          (equivocators_fixed_equivocations_constraint
             IM Hbs index_listing index_listing)))
  split.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
VLSM_incl
  (mk_vlsm
     (composite_vlsm_machine (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM Hbs index_listing index_listing)))
  (mk_vlsm
     (composite_vlsm_machine (equivocator_IM IM)
        (equivocators_no_equivocations_constraint IM
           Hbs)))
message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
VLSM_incl
  (mk_vlsm
     (composite_vlsm_machine 
        (equivocator_IM IM)
        (equivocators_no_equivocations_constraint IM
           Hbs)))
  (mk_vlsm
     (composite_vlsm_machine 
        (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM Hbs index_listing index_listing)))
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
VLSM_incl
  (mk_vlsm
     (composite_vlsm_machine (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM Hbs index_listing index_listing)))
  (mk_vlsm
     (composite_vlsm_machine (equivocator_IM IM)
        (equivocators_no_equivocations_constraint IM
           Hbs))) apply constraint_subsumption_incl.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
input_valid_constraint_subsumption (equivocator_IM IM)
  (equivocators_fixed_equivocations_constraint IM Hbs
     index_listing index_listing)
  (equivocators_no_equivocations_constraint IM Hbs)
    apply preloaded_constraint_subsumption_stronger.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
preloaded_constraint_subsumption (equivocator_IM IM)
  (equivocators_fixed_equivocations_constraint IM Hbs
     index_listing index_listing)
  (equivocators_no_equivocations_constraint IM Hbs)
    apply strong_constraint_subsumption_strongest.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
strong_constraint_subsumption (equivocator_IM IM)
  (equivocators_fixed_equivocations_constraint IM Hbs
     index_listing index_listing)
  (equivocators_no_equivocations_constraint IM Hbs)
    apply strong_constraint_subsumption_all_fixed.
  -message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
VLSM_incl
  (mk_vlsm
     (composite_vlsm_machine (equivocator_IM IM)
        (equivocators_no_equivocations_constraint IM
           Hbs)))
  (mk_vlsm
     (composite_vlsm_machine (equivocator_IM IM)
        (equivocators_fixed_equivocations_constraint
           IM Hbs index_listing index_listing))) apply
      (constraint_subsumption_incl (equivocator_IM IM)
        (equivocators_no_equivocations_constraint IM Hbs)
        (equivocators_fixed_equivocations_constraint IM Hbs index_listing index_listing)).message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
input_valid_constraint_subsumption (equivocator_IM IM)
  (equivocators_no_equivocations_constraint IM Hbs)
  (equivocators_fixed_equivocations_constraint IM Hbs
     index_listing index_listing)
    apply preloaded_constraint_subsumption_stronger.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
preloaded_constraint_subsumption (equivocator_IM IM)
  (equivocators_no_equivocations_constraint IM Hbs)
  (equivocators_fixed_equivocations_constraint IM Hbs
     index_listing index_listing)
    apply strong_constraint_subsumption_strongest.message, index: Type
IndEqDec: EqDecision index
IM: index → VLSM message
Hbs: ∀ i : index, HasBeenSentCapability (IM i)
index_listing: list index
finite_index: Listing index_listing
XE:= equivocators_fixed_equivocations_vlsm IM Hbs
  index_listing index_listing: VLSM message
NE:= equivocators_no_equivocations_vlsm IM Hbs: VLSM message
strong_constraint_subsumption (equivocator_IM IM)
  (equivocators_no_equivocations_constraint IM Hbs)
  (equivocators_fixed_equivocations_constraint IM Hbs
     index_listing index_listing)
    apply strong_constraint_subsumption_fixed_all.
Qed.

End all_equivocating.